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Graphical Representations

Graphical representations in physics involve using graphs to visually represent data or relationships between variables. They help us interpret information more easily and identify patterns or trends.

Imagine you have a map showing different routes to your destination. The map provides you with a visual representation that allows you to see the best path to take. Similarly, graphical representations in physics act as maps that guide us towards understanding complex concepts by presenting information visually.

Related terms

Position-Time Graphs : These graphical representations show an object's position at different points in time, helping us analyze its motion.

Velocity-Time Graphs : These graphs illustrate an object's velocity over time, allowing us to examine changes in speed or direction.

Force-Distance Graphs : By plotting force against distance, these graphs help visualize how force varies as an object moves or interacts with other objects.

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AP Physics 1 - Unit 1 FRQ (Kinematics)

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  • Graphical Representation of Motion

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An Introduction to Motion

When the position of an object changes with time with respect to some reference point, we can say that the object is in motion . The reference point is the point from which the change in position of an object is observed. Graphs offer an easy method to showcase the relationship between two physical quantities.

Importance of Graphs in Motion

Graphs provide us a convenient method to describe some basic information about a variety of events. The nature of the graph can be studied with the help of the slope. The slope is the inclination that the graph makes with the x-axis. It can also be found out by taking the ratio of change in the y-coordinates to change in the x-coordinates. Graphical representation of motion generally uses graphs and with the help of graphs, we can represent the motion of an object. In the graphical representation of motion , we should take the dependent quantity along the y-axis and the independent quantity along the x-axis.

Distance-Time Graph

Position of an object with time can be represented by a distance-time graph . The slope of the distance-time graph gives us the speed of an object. Distance-time graphs can be made by considering two situations, either the object moves with a uniform speed or with non-uniform speed. First, we will consider the case of uniform speed. If an object is moving with a uniform speed, we can say that it covers equal distances in equal time intervals. So, we can say that the distance is directly proportional to the time taken.

Suppose, the object covers 10m in the first 5 seconds, it covers another 10m in the next 5 sec and so on. If we mark these points on the graph, the result that we get would be like this:

Distance-Time Graph for Uniform Motion

Distance - Time Graph for Uniform Motion

Here, we can see a straight line passing from the origin. Thus, for a uniform speed, the graph of distance travelled against time is a straight line. We can also determine the speed of an object from the distance-time graph . The speed of an object can be determined by calculating the slope of the distance-time graph .

We can plot the distance-time graph for accelerated motion as well. Below is the example of the distance-time graph for non-uniform motion. The nature of this graph shows non-linear variation of the distance travelled by the object with time. So, the below graph shows the motion of an object with non-uniform speed.

Distance-Time Graph for Non-Uniform Motion

Distance - Time Graph for Non-Uniform Motion

Displacement-Time Graph

Displacement is the shortest distance between two points. Displacement-time graph gives us the information of velocity of an object. The rate at which displacement varies with time is known as velocity and the slope of the displacement-time graph gives us the velocity of an object. So, if this graph gives us a straight line which is parallel to the x-axis, then we can say that object is at rest.

Velocity-Time Graph

The rate at which velocity varies with time is called acceleration and the slope of this graph gives us the information of acceleration of an object. The variation in velocity with time for an object moving in a straight line can be described by a velocity-time graph .

Velocity-Time Graph

Velocity - Time Graph

Here, time and velocity are represented along the x-axis and y-axis, respectively. The graph shows a straight line which is parallel to the x-axis. That means the object has uniform velocity.

Solved Examples

1. An object is moving along a circular path of radius 7 cm. What is the distance of an object when it completes half revolution.

Ans: Given, radius $r = 7\,cm$

The object is moving along a circular path. We have to calculate the distance travelled by this object when it completes half revolution.

The circumference of the circle is $2\pi r$.

So, here, its distance travelled by it would be $\pi r$.

The distance of an object would be 22 cm when it completes half revolution.

2. A car decreases its speed from $22\,\dfrac{m}{s}$ to $16\,\dfrac{m}{s}$ in 5 sec. Find the acceleration of the car.

Ans: Given, initial speed of the car is $u = 22\,\dfrac{m}{s}$

Final speed of the car $v = 16\,\dfrac{m}{s}$

Time taken $t = 5\sec $

We will use the relation $a = \dfrac{{v - u}}{t}$ to solve this

$\therefore a = \dfrac{{16 - 22}}{5} = - 1.2\,\dfrac{m}{{{s^2}}}$

So, the acceleration of the car is $a = - 1.2\,\dfrac{m}{{{s^2}}}$

Interesting Facts

Displacement can be defined from the area under the velocity-time graph.

The displacement of an object is proportional to the square of time, then we can say that the object moves with uniform acceleration .

A speedometer is a good example of instantaneous speed.

Motion can be defined with the help of graphs. We use graphs to describe the motion of an object. In a line graph, we can represent one physical quantity like distance or velocity with another quantity such as time. The different types of graphs for motion help us to understand the speed, velocity, acceleration, and also behaviour of an object.

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FAQs on Graphical Representation of Motion

1. State the difference between speed and velocity.

Here are a few points which show the difference between speed and velocity.

The speed of the moving object can never be zero and thus the average speed of that object can never be zero. The average velocity of the moving object can be zero when the displacement of the object is zero.

Speed gives us an idea about how much an object has travelled in a certain period of time but in the case of velocity, it is a measure of the rapidity of the motion along with the direction of the object.

Speed is a scalar quantity. So, it gives us information about only magnitude. Velocity is a vector quantity, so we can get information about magnitude and direction.

2. Define positive acceleration and negative acceleration.

If the velocity of an object is increasing with time, then we can say that the acceleration of that object is positive. Final velocity of an object will be greater than the initial velocity of an object in case of positive acceleration. The direction of positive acceleration is the same as the direction of motion of the object.

If the velocity of an object is decreasing with time, then we can say that the acceleration of that object will be negative. The direction of negative acceleration will be opposite to the direction of motion of the object. The initial velocity will be greater than the final velocity in case of negative acceleration. Negative acceleration is also called de-acceleration or retardation.

3. State the difference between distance and displacement.

We define distance as the actual path travelled by the object but displacement is the shortest distance between the two points which an object takes. Distance travelled is not a unique path between two points but the displacement refers to a unique path between two points. The distance travelled gives the information of the type of path followed by the object. Displacement between two points gives information only about the initial and the final positions of the object. Distance never decreases with time for a moving object. It can never be zero. Displacement can decrease with time for a moving object.

Graphs of Motion

Introduction.

Modern mathematical notation is a highly compact way to encode ideas. Equations can easily contain the information equivalent of several sentences. Galileo's description of an object moving with constant speed (perhaps the first application of mathematics to motion) required one definition, four axioms, and six theorems . All of these relationships can now be written in a single equation.

When it comes to depth, nothing beats an equation.

Well, almost nothing. Think back to the previous section on the equations of motion. You should recall that the three (or four) equations presented in that section were only valid for motion with constant acceleration along a straight line. Since, as I rightly pointed out, "no object has ever traveled in a straight line with constant acceleration anywhere in the universe at any time" these equations are only approximately true, only once in a while.

Equations are great for describing idealized situations, but they don't always cut it. Sometimes you need a picture to show what's going on — a mathematical picture called a graph. Graphs are often the best way to convey descriptions of real world events in a compact form. Graphs of motion come in several types depending on which of the kinematic quantities (time, position, velocity, acceleration) are assigned to which axis.

position-time

Let's begin by graphing some examples of motion at a constant velocity. Three different curves are included on the graph to the right, each with an initial position of zero. Note first that the graphs are all straight. (Any kind of line drawn on a graph is called a curve. Even a straight line is called a curve in mathematics.) This is to be expected given the linear nature of the appropriate equation. (The independent variable of a linear function is raised no higher than the first power.)

Compare the position-time equation for constant velocity with the classic slope-intercept equation taught in introductory algebra.

Thus velocity corresponds to slope and initial position to the intercept on the vertical axis (commonly thought of as the "y" axis). Since each of these graphs has its intercept at the origin, each of these objects had the same initial position. This graph could represent a race of some sort where the contestants were all lined up at the starting line (although, at these speeds it must have been a race between tortoises). If it were a race, then the contestants were already moving when the race began, since each curve has a non-zero slope at the start. Note that the initial position being zero does not necessarily imply that the initial velocity is also zero. The height of a curve tells you nothing about its slope.

  • slope is velocity
  • the "y" intercept is the initial position
  • when two curves coincide, the two objects have the same position at that time

In contrast to the previous examples, let's graph the position of an object with a constant, non-zero acceleration starting from rest at the origin. The primary difference between this curve and those on the previous graph is that this curve actually curves. The relation between position and time is quadratic when the acceleration is constant and therefore this curve is a parabola . (The variable of a quadratic function is raised no higher than the second power.)

As an exercise, let's calculate the acceleration of this object from its graph. It intercepts the origin, so its initial position is zero, the example states that the initial velocity is zero, and the graph shows that the object has traveled 9 m in 10 s. These numbers can then be entered into the equation.

When a position-time graph is curved, it is not possible to calculate the velocity from it's slope. Slope is a property of straight lines only. Such an object doesn't have a velocity because it doesn't have a slope. The words "the" and "a" are underlined here to stress the idea that there is no single velocity under these circumstances. The velocity of such an object must be changing. It's accelerating.

  • straight segments imply constant velocity
  • curve segments imply acceleration
  • an object undergoing constant acceleration traces a portion of a parabola

Although our hypothetical object has no single velocity, it still does have an average velocity and a continuous collection of instantaneous velocities. The average velocity of any object can be found by dividing the overall change in position (a.k.a. the displacement) by the change in time.

This is the same as calculating the slope of the straight line connecting the first and last points on the curve as shown in the diagram to the right. In this abstract example, the average velocity of the object was…

Instantaneous velocity is the limit of average velocity as the time interval shrinks to zero.

As the endpoints of the line of average velocity get closer together, they become a better indicator of the actual velocity. When the two points coincide, the line is tangent to the curve. This limit process is represented in the animation to the right.

  • average velocity is the slope of the straight line connecting the endpoints of a curve
  • instantaneous velocity is the slope of the line tangent to a curve at any point

Seven tangents were added to our generic position-time graph in the animation shown above. Note that the slope is zero twice — once at the top of the bump at 3.0 s and again in the bottom of the dent at 6.5 s. (The bump is a local maximum , while the dent is a local minimum . Collectively such points are known as local extrema .) The slope of a horizontal line is zero, meaning that the object was motionless at those times. Since the graph is not flat, the object was only at rest for an instant before it began moving again. Although its position was not changing at that time, its velocity was. This is a notion that many people have difficulty with. It is possible to be accelerating and yet not be moving, but only for an instant.

Note also that the slope is negative in the interval between the bump at 3.0 s and the dent at 6.5 s. Some interpret this as motion in reverse, but is this generally the case? Well, this is an abstract example. It's not accompanied by any text. Graphs contain a lot of information, but without a title or other form of description they have no meaning. What does this graph represent? A person? A car? An elevator? A rhinoceros? An asteroid? A mote of dust? About all we can say is that this object was moving at first, slowed to a stop, reversed direction, stopped again, and then resumed moving in the direction it started with (whatever direction that was). Negative slope does not automatically mean driving backward, or walking left, or falling down. The choice of signs is always arbitrary. About all we can say in general, is that when the slope is negative, the object is traveling in the negative direction.

  • positive slope implies motion in the positive direction
  • negative slope implies motion in the negative direction
  • zero slope implies a state of rest

velocity-time

The most important thing to remember about velocity-time graphs is that they are velocity-time graphs, not position-time graphs. There is something about a line graph that makes people think they're looking at the path of an object. A common beginner's mistake is to look at the graph to the right and think that the the v  = 9.0 m/s line corresponds to an object that is "higher" than the other objects. Don't think like this. It's wrong.

Don't look at these graphs and think of them as a picture of a moving object. Instead, think of them as the record of an object's velocity. In these graphs, higher means faster not farther. The v  = 9.0 m/s line is higher because that object is moving faster than the others.

These particular graphs are all horizontal. The initial velocity of each object is the same as the final velocity is the same as every velocity in between. The velocity of each of these objects is constant during this ten second interval.

In comparison, when the curve on a velocity-time graph is straight but not horizontal, the velocity is changing. The three curves to the right each have a different slope. The graph with the steepest slope experiences the greatest rate of change in velocity. That object has the greatest acceleration. Compare the velocity-time equation for constant acceleration with the classic slope-intercept equation taught in introductory algebra.

You should see that acceleration corresponds to slope and initial velocity to the intercept on the vertical axis. Since each of these graphs has its intercept at the origin, each of these objects was initially at rest. The initial velocity being zero does not mean that the initial position must also be zero, however. This graph tells us nothing about the initial position of these objects. For all we know they could be on different planets.

  • slope is acceleration
  • the "y" intercept is the initial velocity
  • when two curves coincide, the two objects have the same velocity at that time

The curves on the previous graph were all straight lines. A straight line is a curve with constant slope. Since slope is acceleration on a velocity-time graph, each of the objects represented on this graph is moving with a constant acceleration. Were the graphs curved, the acceleration would have been not constant.

  • straight lines imply constant acceleration
  • curved lines imply non-constant acceleration
  • an object undergoing constant acceleration traces a straight line

Since a curved line has no single slope we must decide what we mean when asked for the acceleration of an object. These descriptions follow directly from the definitions of average and instantaneous acceleration. If the average acceleration is desired, draw a line connecting the endpoints of the curve and calculate its slope. If the instantaneous acceleration is desired, take the limit of this slope as the time interval shrinks to zero, that is, take the slope of a tangent.

  • average acceleration is the slope of the straight line connecting the endpoints of a curve
  • instantaneous acceleration is the slope of the line tangent to a curve at any point

Seven tangents were added to our generic velocity-time graph in the animation shown above. Note that the slope is zero twice — once at the top of the bump at 3.0 s and again in the bottom of the dent at 6.5 s. The slope of a horizontal line is zero, meaning that the object stopped accelerating instantaneously at those times. The acceleration might have been zero at those two times, but this does not mean that the object stopped. For that to occur, the curve would have to intercept the horizontal axis. This happened only once — at the start of the graph. At both times when the acceleration was zero, the object was still moving in the positive direction.

You should also notice that the slope was negative from 3.0 s to 6.5 s. During this time the speed was decreasing. This is not true in general, however. Speed decreases whenever the curve returns to the origin. Above the horizontal axis this would be a negative slope, but below it this would be a positive slope. About the only thing one can say about a negative slope on a velocity-time graph is that during such an interval, the velocity is becoming more negative (or less positive, if you prefer).

  • positive slope implies an increase in velocity in the positive direction
  • negative slope implies an increase in velocity in the negative direction
  • zero slope implies motion with constant velocity

In kinematics, there are three quantities: position, velocity, and acceleration. Given a graph of any of these quantities, it is always possible in principle to determine the other two. Acceleration is the time rate of change of velocity, so that can be found from the slope of a tangent to the curve on a velocity-time graph. But how could position be determined? Let's explore some simple examples and then derive the relationship.

Start with the simple velocity-time graph shown to the right. (For the sake of simplicity, let's assume that the initial position is zero.) There are three important intervals on this graph. During each interval, the acceleration is constant as the straight line segments show. When acceleration is constant, the average velocity is just the average of the initial and final values in an interval.

0–4 s: This segment is triangular. The area of a triangle is one-half the base times the height. Essentially, we have just calculated the area of the triangular segment on this graph.

The cumulative distance traveled at the end of this interval is…

4–8 s: This segment is trapezoidal. The area of a trapezoid (or trapezium ) is the average of the two bases times the altitude. Essentially, we have just calculated the area of the trapezoidal segment on this graph.

16 m + 36 m = 52 m

8–10 s: This segment is rectangular. The area of a rectangle is just its height times its width. Essentially, we have just calculated the area of the rectangular segment on this graph.

16 m + 36 m + 20 m = 72 m

I hope by now that you see the trend. The area under each segment is the change in position of the object during that interval. This is true even when the acceleration is not constant.

Anyone who has taken a calculus course should have known this before they read it here (or at least when they read it they should have said, "Oh yeah, I remember that"). The first derivative of position with respect to time is velocity. The derivative of a function is the slope of a line tangent to its curve at a given point. The inverse operation of the derivative is called the integral. The integral of a function is the cumulative area between the curve and the horizontal axis over some interval. This inverse relation between the actions of derivative (slope) and integral (area) is so important that it's called the fundamental theorem of calculus . This means that it's an important relationship. Learn it! It's "fundamental". You haven't seen the last of it.

  • the area under the curve is the change in position

acceleration-time

The acceleration-time graph of any object traveling with a constant velocity is the same. This is true regardless of the velocity of the object. An airplane flying at a constant 270 m/s (600 mph), a sloth walking at a constant 0.4 m/s (1 mph), and a couch potato lying motionless in front of the TV for hours will all have the same acceleration-time graphs — a horizontal line collinear with the horizontal axis. That's because the velocity of each of these objects is constant. They're not accelerating. Their accelerations are zero. As with velocity-time graphs, the important thing to remember is that the height above the horizontal axis doesn't correspond to position or velocity, it corresponds to acceleration .

If you trip and fall on your way to school, your acceleration towards the ground is greater than you'd experience in all but a few high performance cars with the "pedal to the metal". Acceleration and velocity are different quantities. Going fast does not imply accelerating quickly. The two quantities are independent of one another. A large acceleration corresponds to a rapid change in velocity, but it tells you nothing about the values of the velocity itself.

When acceleration is constant, the acceleration-time curve is a horizontal line. The rate of change of acceleration with time is not often discussed, so the slope of the curve on this graph will be ignored for now. If you enjoy knowing the names of things, this quantity is called jerk . On the surface, the only information one can glean from an acceleration-time graph appears to be the acceleration at any given time.

  • slope is jerk
  • the "y" intercept equals the initial acceleration
  • when two curves coincide, the two objects have the same acceleration at that time
  • an object undergoing constant acceleration traces a horizontal line
  • zero slope implies motion with constant acceleration

Acceleration is the rate of change of velocity with time. Transforming a velocity-time graph to an acceleration-time graph means calculating the slope of a line tangent to the curve at any point. (In calculus, this is called finding the derivative.) The reverse process entails calculating the cumulative area under the curve. (In calculus, this is called finding the integral.) This number is then the change of value on a velocity-time graph.

Given an initial velocity of zero (and assuming that down is positive), the final velocity of the person falling in the graph to the right is…

and the final velocity of the accelerating car is…

  • the area under the curve equals the change in velocity

There are more things one can say about acceleration-time graphs, but they are trivial for the most part.

phase space

There is a fourth graph of motion that relates velocity to position. It is as important as the other three types, but it rarely gets any attention below the advanced undergraduate level. Some day I will write something about these graphs called phase space diagrams, but not today.

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2 Kinematics

14 2.8 Graphical Analysis of One-Dimensional Motion

  • Describe a straight-line graph in terms of its slope and y-intercept.
  • Determine average velocity or instantaneous velocity from a graph of position vs. time.
  • Determine average or instantaneous acceleration from a graph of velocity vs. time.
  • Derive a graph of velocity vs. time from a graph of position vs. time.
  • Derive a graph of acceleration vs. time from a graph of velocity vs. time.

A graph, like a picture, is worth a thousand words. Graphs not only contain numerical information; they also reveal relationships between physical quantities. This section uses graphs of displacement, velocity, and acceleration versus time to illustrate one-dimensional kinematics.

Slopes and General Relationships

First note that graphs in this text have perpendicular axes, one horizontal and the other vertical. When two physical quantities are plotted against one another in such a graph, the horizontal axis is usually considered to be an independent variable and the vertical axis a dependent variable . If we call the horizontal axis the[latex]\boldsymbol{x}\text{-axis}[/latex]and the vertical axis the[latex]\boldsymbol{y}\text{-axis}[/latex], as in Figure 1 , a straight-line graph has the general form

Here[latex]\boldsymbol{m}[/latex]is the slope , defined to be the rise divided by the run (as seen in the figure) of the straight line. The letter[latex]\boldsymbol{b}[/latex]is used for the y -intercept , which is the point at which the line crosses the vertical axis.

Graph of a straight-line sloping up at about 40 degrees.

Graph of Displacement vs. Time ( a = 0, so v is constant)

Time is usually an independent variable that other quantities, such as displacement, depend upon. A graph of displacement versus time would, thus, have[latex]\boldsymbol{x}[/latex]on the vertical axis and[latex]\boldsymbol{t}[/latex]on the horizontal axis. Figure 2 is just such a straight-line graph. It shows a graph of displacement versus time for a jet-powered car on a very flat dry lake bed in Nevada .

Line graph of jet car displacement in meters versus time in seconds. The line is straight with a positive slope. The y intercept is four hundred meters. The total change in time is eight point zero seconds. The initial position is four hundred meters. The final position is two thousand meters.

Using the relationship between dependent and independent variables, we see that the slope in the graph above is average velocity[latex]\boldsymbol{\bar{v}}[/latex]and the intercept is displacement at time zero—that is,[latex]\boldsymbol{x_0}.[/latex]Substituting these symbols into[latex]\boldsymbol{y}=\boldsymbol{mx+b}[/latex]gives

Thus a graph of displacement versus time gives a general relationship among displacement, velocity, and time, as well as giving detailed numerical information about a specific situation.

THE SLOPE OF X VS. T

The slope of the graph of displacement [latex]\boldsymbol{x}[/latex]vs. time[latex]\boldsymbol{t}[/latex]is velocity[latex]\boldsymbol{v}.[/latex]

Notice that this equation is the same as that derived algebraically from other motion equations in Chapter 2.5 Motion Equations for Constant Acceleration in One Dimension .

From the figure we can see that the car has a displacement of 25 m at 0.50 s and 2000 m at 6.40 s. Its displacement at other times can be read from the graph; furthermore, information about its velocity and acceleration can also be obtained from the graph.

Example 1: Determining Average Velocity from a Graph of Displacement versus Time: Jet Car

Find the average velocity of the car whose position is graphed in Figure 2 .

The slope of a graph of[latex]\boldsymbol{x}[/latex]vs.[latex]\boldsymbol{t}[/latex]is average velocity, since slope equals rise over run. In this case, rise = change in displacement and run = change in time, so that

Since the slope is constant here, any two points on the graph can be used to find the slope. (Generally speaking, it is most accurate to use two widely separated points on the straight line. This is because any error in reading data from the graph is proportionally smaller if the interval is larger.)

1. Choose two points on the line. In this case, we choose the points labeled on the graph: (6.4 s, 2000 m) and (0.50 s, 525 m). (Note, however, that you could choose any two points.)

2. Substitute the[latex]\boldsymbol{x}[/latex]and[latex]\boldsymbol{t}[/latex]values of the chosen points into the equation. Remember in calculating change[latex]\boldsymbol{(\Delta)}[/latex]we always use final value minus initial value.

This is an impressively large land speed (900 km/h, or about 560 mi/h): much greater than the typical highway speed limit of 60 mi/h (27 m/s or 96 km/h), but considerably shy of the record of 343 m/s (1234 km/h or 766 mi/h) set in 1997.

Graphs of Motion when α is constant but α≠0

The graphs in Figure 3 below represent the motion of the jet-powered car as it accelerates toward its top speed, but only during the time when its acceleration is constant. Time starts at zero for this motion (as if measured with a stopwatch), and the displacement and velocity are initially 200 m and 15 m/s, respectively.

Three line graphs. First is a line graph of displacement over time. Line has a positive slope that increases with time. Second line graph is of velocity over time. Line is straight with a positive slope. Third line graph is of acceleration over time. Line is straight and horizontal, indicating constant acceleration.

The graph of displacement versus time in Figure 3 (a) is a curve rather than a straight line. The slope of the curve becomes steeper as time progresses, showing that the velocity is increasing over time. The slope at any point on a displacement-versus-time graph is the instantaneous velocity at that point. It is found by drawing a straight line tangent to the curve at the point of interest and taking the slope of this straight line. Tangent lines are shown for two points in Figure 3 (a). If this is done at every point on the curve and the values are plotted against time, then the graph of velocity versus time shown in Figure 3 (b) is obtained. Furthermore, the slope of the graph of velocity versus time is acceleration, which is shown in Figure 3 (c).

Example 2: Determining Instantaneous Velocity from the Slope at a Point: Jet Car

Calculate the velocity of the jet car at a time of 25 s by finding the slope of the[latex]\boldsymbol{x}[/latex]vs.[latex]\boldsymbol{t}[/latex]graph in the graph below.

A graph of displacement versus time for a jet car. The x axis for time runs from zero to thirty five seconds. The y axis for displacement runs from zero to three thousand meters. The curve depicting displacement is concave up. The slope of the curve increases over time. Slope equals velocity v. There are two points on the curve, labeled, P and Q. P is located at time equals ten seconds. Q is located and time equals twenty-five seconds. A line tangent to P at ten seconds is drawn and has a slope delta x sub P over delta t sub p. A line tangent to Q at twenty five seconds is drawn and has a slope equal to delta x sub q over delta t sub q. Select coordinates are given in a table and consist of the following: time zero seconds displacement two hundred meters; time five seconds displacement three hundred thirty eight meters; time ten seconds displacement six hundred meters; time fifteen seconds displacement nine hundred eighty eight meters. Time twenty seconds displacement one thousand five hundred meters; time twenty five seconds displacement two thousand one hundred thirty eight meters; time thirty seconds displacement two thousand nine hundred meters.

The slope of a curve at a point is equal to the slope of a straight line tangent to the curve at that point. This principle is illustrated in Figure 5 , where Q is the point at[latex]\boldsymbol{t=25\textbf{ s}}.[/latex]

1. Find the tangent line to the curve at[latex]\boldsymbol{t=25\textbf{ s}}.[/latex]

2. Determine the endpoints of the tangent. These correspond to a position of 1300 m at time 19 s and a position of 3120 m at time 32 s.

3. Plug these endpoints into the equation to solve for the slope,[latex]\boldsymbol{v}.[/latex]

This is the value given in this figure’s table for[latex]\boldsymbol{v}[/latex]at[latex]\boldsymbol{t=25\textbf{ s}}.[/latex]The value of 140 m/s for[latex]\boldsymbol{v_Q}[/latex]is plotted in Figure 5 . The entire graph of[latex]\boldsymbol{v}[/latex]vs.[latex]\boldsymbol{t}[/latex]can be obtained in this fashion.

Carrying this one step further, we note that the slope of a velocity versus time graph is acceleration. Slope is rise divided by run; on a[latex]\boldsymbol{v}[/latex]vs.[latex]\boldsymbol{t}[/latex]graph, rise = change in velocity[latex]\boldsymbol{\Delta{v}}[/latex]and run = change in time[latex]\boldsymbol{\Delta{t}}.[/latex]

THE SLOPE OF V VS. T

The slope of a graph of velocity[latex]\boldsymbol{v}[/latex]vs. time[latex]\boldsymbol{t}[/latex]is acceleration[latex]\boldsymbol{a}.[/latex]

Since the velocity versus time graph in Figure 3 (b) is a straight line, its slope is the same everywhere, implying that acceleration is constant. Acceleration versus time is graphed in Figure 3 (c).

Additional general information can be obtained from Figure 5 and the expression for a straight line,[latex]\boldsymbol{y=mx+b}.[/latex]

In this case, the vertical axis[latex]\boldsymbol{y}[/latex]is[latex]\textbf{V}[/latex], the intercept[latex]\boldsymbol{b}[/latex]is[latex]\boldsymbol{v_0},[/latex]the slope[latex]\boldsymbol{m}[/latex]is[latex]\boldsymbol{a},[/latex]and the horizontal axis[latex]\boldsymbol{x}[/latex] is [latex]\boldsymbol{t}.[/latex]Substituting these symbols yields

A general relationship for velocity, acceleration, and time has again been obtained from a graph. Notice that this equation was also derived algebraically from other motion equations in Chapter 2.5 Motion Equations for Constant Acceleration in One Dimension .

It is not accidental that the same equations are obtained by graphical analysis as by algebraic techniques. In fact, an important way to discover physical relationships is to measure various physical quantities and then make graphs of one quantity against another to see if they are correlated in any way. Correlations imply physical relationships and might be shown by smooth graphs such as those above. From such graphs, mathematical relationships can sometimes be postulated. Further experiments are then performed to determine the validity of the hypothesized relationships.

Graphs of Motion Where Acceleration is Not Constant

Now consider the motion of the jet car as it goes from 165 m/s to its top velocity of 250 m/s, graphed in Figure 6 . Time again starts at zero, and the initial displacement and velocity are 2900 m and 165 m/s, respectively. (These were the final displacement and velocity of the car in the motion graphed in Figure 3 .) Acceleration gradually decreases from[latex]\boldsymbol{5.0\textbf{ m/s}^2}[/latex]to zero when the car hits 250 m/s. The slope of the[latex]\boldsymbol{x}[/latex]vs.[latex]\boldsymbol{t}[/latex]graph increases until[latex]\boldsymbol{t=55\textbf{ s}},[/latex]after which time the slope is constant. Similarly, velocity increases until 55 s and then becomes constant, since acceleration decreases to zero at 55 s and remains zero afterward.

Three line graphs of jet car displacement, velocity, and acceleration, respectively. First line graph is of position over time. Line is straight with a positive slope. Second line graph is of velocity over time. Line graph has a positive slope that decreases over time and flattens out at the end. Third line graph is of acceleration over time. Line has a negative slope that increases over time until it flattens out at the end. The line is not smooth, but has several kinks.

Example 3: Calculating Acceleration from a Graph of Velocity versus Time

Calculate the acceleration of the jet car at a time of 25 s by finding the slope of the[latex]\boldsymbol{v}[/latex]vs.[latex]\boldsymbol{t}[/latex]graph in Figure 6 (b).

The slope of the curve at[latex]\boldsymbol{t=25\textbf{ s}}[/latex]is equal to the slope of the line tangent at that point, as illustrated in Figure 6 (b).

Determine endpoints of the tangent line from the figure, and then plug them into the equation to solve for slope,[latex]\boldsymbol{a}.[/latex]

Note that this value for[latex]\boldsymbol{a}[/latex]is consistent with the value plotted in Figure 6 (c) at[latex]\boldsymbol{t=25\textbf{ s}}.[/latex]

A graph of displacement versus time can be used to generate a graph of velocity versus time, and a graph of velocity versus time can be used to generate a graph of acceleration versus time. We do this by finding the slope of the graphs at every point. If the graph is linear (i.e., a line with a constant slope), it is easy to find the slope at any point and you have the slope for every point. Graphical analysis of motion can be used to describe both specific and general characteristics of kinematics. Graphs can also be used for other topics in physics. An important aspect of exploring physical relationships is to graph them and look for underlying relationships.

Check Your Understanding

1: A graph of velocity vs. time of a ship coming into a harbor is shown below. (a) Describe the motion of the ship based on the graph. (b)What would a graph of the ship’s acceleration look like?

Line graph of velocity versus time. The line has three legs. The first leg is flat. The second leg has a negative slope. The third leg also has a negative slope, but the slope is not as negative as the second leg.

Section Summary

  • Graphs of motion can be used to analyze motion.
  • Graphical solutions yield identical solutions to mathematical methods for deriving motion equations.
  • The slope of a graph of displacement[latex]\boldsymbol{x}[/latex]vs. time[latex]\boldsymbol{t}[/latex]is velocity[latex]\boldsymbol{v}.[/latex]
  • The slope of a graph of velocity[latex]\boldsymbol{v}[/latex]vs. time[latex]\boldsymbol{t}[/latex]graph is acceleration[latex]\boldsymbol{a}.[/latex]
  • Average velocity, instantaneous velocity, and acceleration can all be obtained by analyzing graphs.

Conceptual Questions

1: (a) Explain how you can use the graph of position versus time in Figure 8 to describe the change in velocity over time. Identify (b) the time ([latex]\boldsymbol{t_a, t_b,t_c,t_d,}\text{ or }\boldsymbol{t_e}[/latex]) at which the instantaneous velocity is greatest, (c) the time at which it is zero, and (d) the time at which it is negative.

Line graph of position versus time with 5 points labeled: a, b, c, d, and e. The slope of the line changes. It begins with a positive slope that decreases over time until around point d, where it is flat. It then has a slightly negative slope.

2: (a) Sketch a graph of velocity versus time corresponding to the graph of displacement versus time given in Figure 9 . (b) Identify the time or times ([latex]\boldsymbol{t_a,t_b,t_c},[/latex]etc.) at which the instantaneous velocity is greatest. (c) At which times is it zero? (d) At which times is it negative?

Line graph of position over time with 12 points labeled a through l. Line has a negative slope from a to c, where it turns and has a positive slope till point e. It turns again and has a negative slope till point g. The slope then increases again till l, where it flattens out.

3: (a) Explain how you can determine the acceleration over time from a velocity versus time graph such as the one in Figure 10 . (b) Based on the graph, how does acceleration change over time?

Line graph of velocity over time with two points labeled. Point P is at v 1 t 1. Point Q is at v 2 t 2. The line has a positive slope that increases over time.

4: (a) Sketch a graph of acceleration versus time corresponding to the graph of velocity versus time given in Figure 11 . (b) Identify the time or times ([latex]\boldsymbol{t_a,t_b,t_c},[/latex]etc.) at which the acceleration is greatest. (c) At which times is it zero? (d) At which times is it negative?

Line graph of velocity over time with 12 points labeled a through l. The line has a positive slope from a at the origin to d where it slopes downward to e, and then back upward to h. It then slopes back down to point l at v equals 0.

5: Consider the velocity vs. time graph of a person in an elevator shown in Figure 12 . Suppose the elevator is initially at rest. It then accelerates for 3 seconds, maintains that velocity for 15 seconds, then decelerates for 5 seconds until it stops. The acceleration for the entire trip is not constant so we cannot use the equations of motion from Chapter 2.5 Motion Equations for Constant Acceleration in One Dimension for the complete trip. (We could, however, use them in the three individual sections where acceleration is a constant.) Sketch graphs of (a) position vs. time and (b) acceleration vs. time for this trip.

Line graph of velocity versus time. Line begins at the origin and has a positive slope until it reaches 3 meters per second at 3 seconds. The slope is then zero until 18 seconds, where it becomes negative until the line reaches a velocity of 0 at 23 seconds.

6: A cylinder is given a push and then rolls up an inclined plane. If the origin is the starting point, sketch the position, velocity, and acceleration of the cylinder vs. time as it goes up and then down the plane.

Problems & Exercises

Note: There is always uncertainty in numbers taken from graphs. If your answers differ from expected values, examine them to see if they are within data extraction uncertainties estimated by you.

1: (a) By taking the slope of the curve in Figure 13 , verify that the velocity of the jet car is 115 m/s at[latex]\boldsymbol{t=20\textbf{ s}}.[/latex](b) By taking the slope of the curve at any point in Figure 14 , verify that the jet car’s acceleration is[latex]\boldsymbol{5.0\textbf{ m/s}^2}.[/latex]

Line graph of position over time. Line has positive slope that increases over time.

2: Using approximate values, calculate the slope of the curve in Figure 15 to verify that the velocity at[latex]\boldsymbol{t=10.0\textbf{ s}}[/latex]is 0.208 m/s. Assume all values are known to 3 significant figures.

Line graph of position versus time. Line is straight with a positive slope.

3: Using approximate values, calculate the slope of the curve in Figure 15 to verify that the velocity at[latex]\boldsymbol{t=30.0\textbf{ s}}[/latex]is 0.238 m/s. Assume all values are known to 3 significant figures.

4: By taking the slope of the curve in Figure 16 , verify that the acceleration is[latex]\boldsymbol{3.2\textbf{ m/s}^2}[/latex]at[latex]\boldsymbol{t=10\textbf{ s}}.[/latex]

Line graph of velocity versus time. Line has a positive slope that decreases over time until the line flattens out.

5: Construct the displacement graph for the subway shuttle train as shown in Chapter 2.4 Figure 7 (a). Your graph should show the position of the train, in kilometers, from t = 0 to 20 s. You will need to use the information on acceleration and velocity given in the examples for this figure.

6: (a) Take the slope of the curve in Figure 17 to find the jogger’s velocity at[latex]\boldsymbol{t=2.5\textbf{ s}}.[/latex](b) Repeat at 7.5 s. These values must be consistent with the graph in Figure 18 .

Line graph of position over time. Line begins sloping upward, then kinks back down, then kinks back upward again.

7: A graph of[latex]\boldsymbol{v(t)}[/latex]is shown for a world-class track sprinter in a 100-m race. (See Figure 20 ). (a) What is his average velocity for the first 4 s? (b) What is his instantaneous velocity at[latex]\boldsymbol{t=5\textbf{ s}}?[/latex](c) What is his average acceleration between 0 and 4 s? (d) What is his time for the race?

Line graph of velocity versus time. The line has two legs. The first has a constant positive slope. The second is flat, with a slope of 0.

8: Figure 21 shows the displacement graph for a particle for 5 s. Draw the corresponding velocity and acceleration graphs.

Line graph of position versus time. The line has 4 legs. The first leg has a positive slope. The second leg has a negative slope. The third has a slope of 0. The fourth has a positive slope.

1: (a) The ship moves at constant velocity and then begins to decelerate at a constant rate. At some point, its deceleration rate decreases. It maintains this lower deceleration rate until it stops moving.

(b) A graph of acceleration vs. time would show zero acceleration in the first leg, large and constant negative acceleration in the second leg, and constant negative acceleration.

A line graph of acceleration versus time. There are three legs of the graph. All three legs are flat and straight. The first leg shows constant acceleration of 0. The second leg shows a constant negative acceleration. The third leg shows a constant negative acceleration that is not as negative as the second leg.

(a)[latex]\boldsymbol{115\textbf{ m/s}}[/latex]

(b)[latex]\boldsymbol{5.0\textbf{ m/s}^2}[/latex]

[latex]\boldsymbol{v=}[/latex][latex]\boldsymbol{\frac{(11.7\:-\:6.95)\times10^3\textbf{ m}}{(40.0\:-\:20.0)\textbf{ s}}}[/latex][latex]\boldsymbol{=238\textbf{ m/s}}[/latex]

Line graph of position versus time. Line begins with a slight positive slope. It then kinks to a much greater positive slope.

(c)[latex]\boldsymbol{3\textbf{ m/s}^2}[/latex]

College Physics chapters 1-17 Copyright © August 22, 2016 by OpenStax is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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  • Graphical Representation of Motion

Graphical Representation makes it simpler for us to understand data. When analyzing motion, graphs representing values of various parameters of motion make it simpler to solve problems. Let us understand the concept of motion and the other entities related to it using the graphical method.

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Using a graph for a pictorial representation of two sets of data is called a graphical representation of data . One entity is represented on the x-axis of the graph while the other is represented on the y-axis. Out of the two entities, one is a dependent set of variables while the other is independent an independent set of variables.

We use line graphs to describe the motion of an object. This graph shows the dependency of a physical quantity speed or distance on another quantity, for example, time.

Browse more Topics under Motion

  • Introduction to Motion and its Parameters
  • Equations of Motion
  • Uniform Circular Motion

Distance Time Graph

The distance-time graph determines the change in the position of the object. The speed of the object as well can be determined using the line graph. Here the time lies on the x-axis while the distance on the y-axis. Remember, the line graph of uniform motion is always a straight line .

Why? Because as the definition goes, uniform motion is when an object covers the equal amount of distance at equal intervals of time.  Hence the straight line. While the graph of a non-uniform motion is a curved graph.

Velocity and  Time Graph

graphical representation physics definition

A velocity-time graph is also a straight line. Here the time is on the x-axis while the velocity is on the y-axis. The product of time and velocity gives the displacement of an object moving at a uniform speed. The velocity of time and graph of a velocity that changes uniformly is a straight line. We can use this graph to calculate the acceleration of the object.

Acceleration =(Change in velocity)/time

For calculating acceleration draw a perpendicular on the x-axis from the graph point as shown in the figure. Here the acceleration will be equal to the slope of the velocity-time graph. Distance travelled will be equal to the area of the triangle, Therefore,

Distance traveled= (Base × Height)/2

Just like in the distance-time graph, when the velocity is non-uniform the velocity-time graph is a curved line.

Solved Examples for You

Question: The graph shows position as a function of time for an object moving along a straight line. During which time(s) is the object at rest?

  • 0.5 seconds
  • From 1 to 2 seconds
  • 2.5 seconds

Graphical Representation of Motion

Solution: Option B. Slope of the curve under the position-time graph gives the instantaneous velocity of the object. The slope of the curve is zero only in the time interval 1 < t < 2 s. Thus the object is at rest (or velocity is zero) only from 1 to 2 s. Hence option B is correct.

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Uniform Motion

PhysicsGoeasy

  • March 24, 2023
  • Kinematics , Mechanics

Table of Contents

Introduction

Uniform motion plays a significant role in understanding the movement of objects in our everyday lives. In this article, we will learn about the concept of uniform motion in physics. We will also learn about its graphical representation along with the initial instant of time. In the end, I will discuss some of the frequently asked questions about this concept.

We will also provide a quiz at the end of the article where you can check your understanding of uniform motion in physics.

Understanding Uniform Motion:

Uniform motion definition:.

Uniform motion is defined as the motion of a particle whose coordinate (position) is a linear function of time.

The equation for this type of motion is $x = vt + b$, where $v$ and $b$ are constants. Any motion that does not fit this definition is considered variable motion .

Equation of Motion:

The equation $x = vt + b$, represents uniform motion, where $x$ is the coordinate of the particle, $v$ is the constant velocity, $t$ is the time, and $b$ is the initial position.

Graphical Representation of Uniform Motion:

Importance of graphs.

Illustrations, such as graphs, are crucial for understanding complex concepts in physics, including uniform and variable motion. They provide a visual representation of the motion, allowing for easier interpretation and analysis of the motion’s characteristics.

Visualizing Motion – Plotting a Graph:

The motion equation can be visually represented by plotting a graph. This is done by constructing a coordinate system with time (t) on the x-axis and the particle’s coordinate (x) on the y-axis.

To better understand this, consider the data from our article on Rectilinear Motion . The table given below shows the measurements of the car’s position in relation to the markers at regular intervals of time.

By plotting the values of the variables from the table on both axes, we proceed to create perpendicular lines from the axes at these points. The intersection of these perpendicular lines forms a series of points.

Drawing a smooth line connecting these points results in a motion graph. As the coordinate in uniform motion is a linear function of time, the resulting graph appears as a straight line as shown below in the figure.

Uniform Motion

The Initial Instant of Time:

The initial instant of time is the point at which an experiment or observation begins. It is not necessarily the beginning of the motion but rather the starting point of the study.

About Initial Coordinate:

The initial coordinate, denoted by $x_0$, is the distance from the moving particle to the origin of coordinates at the initial instant of time. This can be determined by setting $t = 0$ in the equation of motion, which gives $x_0 = b$.

Example of uniform motion

List of few other examples of uniform motion.

  • The revolution or orbital motion of the Earth around the Sun. 
  • Earth’s rotation about its axis
  • A man-made satellite orbiting the Earth.
  • Walking at a steady speed.
  • The hour hand of the clock 
  • A car traveling at a constant speed along a straight, flat route.
  • A sewing machine’s vibrating spring.

Challenges and Limitations:

Although the uniform motion is a fundamental concept in physics, real-world applications often involve variable motion due to factors such as friction, air resistance, and changing forces. Therefore, uniform motion serves as a basis for understanding more complex motion scenarios.

Frequently asked questions

1. what is the difference between uniform and non-uniform motion with examples.

Uniform motion occurs when an object moves at a constant speed in a straight line. In this case, the object covers equal distances in equal intervals of time. For example, a car traveling at a constant speed of 60 km/h on a straight highway exhibits uniform motion.

Non-uniform motion , on the other hand, occurs when an object’s speed, direction, or both change over time. The object does not cover equal distances in equal intervals of time. For example, a roller coaster with varying speeds and changing directions demonstrates non-uniform motion.

2. What is the purpose of uniform motion in physics, and how is it used in everyday life?

Uniform motion is a fundamental concept in physics that helps us understand the motion of objects in the real world. It forms the basis for understanding more complex motion patterns and phenomena. In everyday life, uniform motion can be seen in various situations, such as a person jogging at a constant pace around a track or a conveyor belt moving at a steady speed in a factory.

3. How can one maintain uniform motion?

To maintain uniform motion, an object must maintain a constant speed and move in a straight line. This can be achieved by ensuring that no external forces act upon the object, or that the net force acting on the object is zero. In practice, this can be difficult due to factors such as friction, air resistance, and changes in terrain or environment.

4. What does the path of an object look like when it’s in uniform motion?

When an object is in uniform motion, its path is a straight line. This is because the object moves at a constant speed without changing direction. The graphical representation of uniform motion, with time on the horizontal axis and position on the vertical axis, would also appear as a straight line.

5. Does a body subjected to uniform motion always move in a straight line?

Yes, a body subjected to uniform motion always moves in a straight line. This is because uniform motion is characterized by constant speed and unchanging direction. Any deviation from a straight-line path would indicate a change in direction, which would mean the motion is no longer uniform.

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7 Vector Addition and Subtraction: Graphical Methods

[latexpage]

Learning Objectives

  • Understand the rules of vector addition, subtraction, and multiplication.
  • Apply graphical methods of vector addition and subtraction to determine the displacement of moving objects.

Some Hawaiian Islands like Kauai Oahu, Molokai, Lanai, Maui, Kahoolawe, and Hawaii are shown. On the scale map of Hawaiian Islands the path of a journey is shown moving from Hawaii to Molokai. The path of the journey is turning at different angles and finally reaching its destination. The displacement of the journey is shown with the help of a straight line connecting its starting point and the destination.

Vectors in Two Dimensions

A vector is a quantity that has magnitude and direction. Displacement, velocity, acceleration, and force, for example, are all vectors. In one-dimensional, or straight-line, motion, the direction of a vector can be given simply by a plus or minus sign. In two dimensions (2-d), however, we specify the direction of a vector relative to some reference frame (i.e., coordinate system), using an arrow having length proportional to the vector’s magnitude and pointing in the direction of the vector.

(Figure) shows such a graphical representation of a vector , using as an example the total displacement for the person walking in a city considered in Kinematics in Two Dimensions: An Introduction . We shall use the notation that a boldface symbol, such as \(\text{D}\), stands for a vector. Its magnitude is represented by the symbol in italics, \(D\), and its direction by \(\theta \).

In this text, we will represent a vector with a boldface variable. For example, we will represent the quantity force with the vector \(\text{F}\), which has both magnitude and direction. The magnitude of the vector will be represented by a variable in italics, such as \(F\), and the direction of the variable will be given by an angle \(\theta \).

A graph is shown. On the axes the scale is set to one block is equal to one unit. A helicopter starts moving from the origin at an angle of twenty nine point one degrees above the x axis. The current position of the helicopter is ten point three blocks along its line of motion. The destination of the helicopter is the point which is nine blocks in the positive x direction and five blocks in the positive y direction. The positive direction of the x axis is east and the positive direction of the y axis is north.

Vector Addition: Head-to-Tail Method

The head-to-tail method is a graphical way to add vectors, described in (Figure) below and in the steps following. The tail of the vector is the starting point of the vector, and the head (or tip) of a vector is the final, pointed end of the arrow.

In part a, a vector of magnitude of nine units and making an angle of theta is equal to zero degrees is drawn from the origin and along the positive direction of x axis. In part b a vector of magnitude of nine units and making an angle of theta is equal to zero degree is drawn from the origin and along the positive direction of x axis. Then a vertical arrow from the head of the horizontal arrow is drawn. In part c a vector D of magnitude ten point three is drawn from the tail of the horizontal vector at an angle theta is equal to twenty nine point one degrees from the positive direction of x axis. The head of the vector D meets the head of the vertical vector. A scale is shown parallel to the vector D to measure its length. Also a protractor is shown to measure the inclination of the vectorD.

Step 1. Draw an arrow to represent the first vector (9 blocks to the east) using a ruler and protractor .

In part a, a vector of magnitude of nine units and making an angle theta is equal to zero degree is drawn from the origin and along the positive direction of x axis.

Step 2. Now draw an arrow to represent the second vector (5 blocks to the north). Place the tail of the second vector at the head of the first vector .

In part b, a vector of magnitude of nine units and making an angle theta is equal to zero degree is drawn from the origin and along the positive direction of x axis. Then a vertical vector from the head of the horizontal vector is drawn.

Step 3. If there are more than two vectors, continue this process for each vector to be added. Note that in our example, we have only two vectors, so we have finished placing arrows tip to tail .

Step 4. Draw an arrow from the tail of the first vector to the head of the last vector . This is the resultant , or the sum, of the other vectors.

In part c, a vector D of magnitude ten point three is drawn from the tail of the horizontal vector at an angle theta is equal to twenty nine point one degrees from the positive direction of the x axis. The head of the vector D meets the head of the vertical vector. A scale is shown parallel to the vector D to measure its length. Also a protractor is shown to measure the inclination of the vector D.

Step 5. To get the magnitude of the resultant, measure its length with a ruler. (Note that in most calculations, we will use the Pythagorean theorem to determine this length.)

Step 6. To get the direction of the resultant, measure the angle it makes with the reference frame using a protractor. (Note that in most calculations, we will use trigonometric relationships to determine this angle.)

The graphical addition of vectors is limited in accuracy only by the precision with which the drawings can be made and the precision of the measuring tools. It is valid for any number of vectors.

Use the graphical technique for adding vectors to find the total displacement of a person who walks the following three paths (displacements) on a flat field. First, she walks 25.0 m in a direction \(\text{49.0º}\) north of east. Then, she walks 23.0 m heading \(\text{15.0º}\) north of east. Finally, she turns and walks 32.0 m in a direction 68.0° south of east.

Represent each displacement vector graphically with an arrow, labeling the first \(\text{A}\), the second \(\text{B}\), and the third \(\text{C}\), making the lengths proportional to the distance and the directions as specified relative to an east-west line. The head-to-tail method outlined above will give a way to determine the magnitude and direction of the resultant displacement, denoted \(\mathbf{\text{R}}\).

(1) Draw the three displacement vectors.

On the graph a vector of magnitude twenty three meters and inclined above the x axis at an angle theta-b equal to fifteen degrees is shown. This vector is labeled as B.

(2) Place the vectors head to tail retaining both their initial magnitude and direction.

In this figure a vector A with a positive slope is drawn from the origin. Then from the head of the vector A another vector B with positive slope is drawn and then another vector C with negative slope from the head of the vector B is drawn which cuts the x axis.

(3) Draw the resultant vector, \(\text{R}\).

In this figure a vector A with a positive slope is drawn from the origin. Then from the head of the vector A another vector B with positive slope is drawn and then another vector C with negative slope from the head of the vector B is drawn which cuts the x axis. From the tail of the vector A a vector R of magnitude of fifty point zero meters and with negative slope of seven degrees is drawn. The head of this vector R meets the head of the vector C. The vector R is known as the resultant vector.

(4) Use a ruler to measure the magnitude of \(\mathbf{\text{R}}\), and a protractor to measure the direction of \(\text{R}\). While the direction of the vector can be specified in many ways, the easiest way is to measure the angle between the vector and the nearest horizontal or vertical axis. Since the resultant vector is south of the eastward pointing axis, we flip the protractor upside down and measure the angle between the eastward axis and the vector.

In this figure a vector A with a positive slope is drawn from the origin. Then from the head of the vector A another vector B with positive slope is drawn and then another vector C with negative slope from the head of the vector B is drawn which cuts the x axis. From the tail of the vector A a vector R of magnitude of fifty meter and with negative slope of seven degrees is drawn. The head of this vector R meets the head of the vector C. The vector R is known as the resultant vector. A ruler is placed along the vector R to measure it. Also there is a protractor to measure the angle.

In this case, the total displacement \(\mathbf{\text{R}}\) is seen to have a magnitude of 50.0 m and to lie in a direction \(7.0º\) south of east. By using its magnitude and direction, this vector can be expressed as \(R=\text{50.0 m}\) and \(\theta =7\text{.}\text{0º}\) south of east.

The head-to-tail graphical method of vector addition works for any number of vectors. It is also important to note that the resultant is independent of the order in which the vectors are added. Therefore, we could add the vectors in any order as illustrated in (Figure) and we will still get the same solution.

In this figure a vector C with a negative slope is drawn from the origin. Then from the head of the vector C another vector A with positive slope is drawn and then another vector B with negative slope from the head of the vector A is drawn. From the tail of the vector C a vector R of magnitude of fifty point zero meters and with negative slope of seven degrees is drawn. The head of this vector R meets the head of the vector B. The vector R is known as the resultant vector.

Here, we see that when the same vectors are added in a different order, the result is the same. This characteristic is true in every case and is an important characteristic of vectors. Vector addition is commutative . Vectors can be added in any order.

(This is true for the addition of ordinary numbers as well—you get the same result whether you add \(\mathbf{\text{2}}+\mathbf{\text{3}}\) or \(\mathbf{\text{3}}+\mathbf{\text{2}}\), for example).

Vector Subtraction

Vector subtraction is a straightforward extension of vector addition. To define subtraction (say we want to subtract \(\mathbf{\text{B}}\) from \(\mathbf{\text{A}}\) , written \(\mathbf{\text{A}}–\mathbf{\text{B}}\) , we must first define what we mean by subtraction. The negative of a vector \(\mathbf{\text{B}}\) is defined to be \(\mathbf{\text{–B}}\); that is, graphically the negative of any vector has the same magnitude but the opposite direction , as shown in (Figure) . In other words, \(\mathbf{\text{B}}\) has the same length as \(\mathbf{\text{–B}}\), but points in the opposite direction. Essentially, we just flip the vector so it points in the opposite direction.

Two vectors are shown. One of the vectors is labeled as vector in north east direction. The other vector is of the same magnitude and is in the opposite direction to that of vector B. This vector is denoted as negative B.

The subtraction of vector \(\mathbf{\text{B}}\) from vector \(\mathbf{\text{A}}\) is then simply defined to be the addition of \(\mathbf{\text{–B}}\) to \(\mathbf{\text{A}}\). Note that vector subtraction is the addition of a negative vector. The order of subtraction does not affect the results.

This is analogous to the subtraction of scalars (where, for example, \(\text{5 – 2 = 5 + }\left(\text{–2}\right)\)). Again, the result is independent of the order in which the subtraction is made. When vectors are subtracted graphically, the techniques outlined above are used, as the following example illustrates.

A woman sailing a boat at night is following directions to a dock. The instructions read to first sail 27.5 m in a direction \(\text{66.0º}\) north of east from her current location, and then travel 30.0 m in a direction \(\text{112º}\) north of east (or \(\text{22.0º}\) west of north). If the woman makes a mistake and travels in the opposite direction for the second leg of the trip, where will she end up? Compare this location with the location of the dock.

A vector of magnitude twenty seven point five meters is shown. It is inclined to the horizontal at an angle of sixty six degrees. Another vector of magnitude thirty point zero meters is shown. It is inclined to the horizontal at an angle of one hundred and twelve degrees.

We can represent the first leg of the trip with a vector \(\mathbf{\text{A}}\), and the second leg of the trip with a vector \(\mathbf{\text{B}}\). The dock is located at a location \(\mathbf{\text{A}}+\mathbf{\text{B}}\). If the woman mistakenly travels in the opposite direction for the second leg of the journey, she will travel a distance \(B\) (30.0 m) in the direction \(180º–112º=68º\) south of east. We represent this as \(\mathbf{\text{–B}}\), as shown below. The vector \(\mathbf{\text{–B}}\) has the same magnitude as \(\mathbf{\text{B}}\) but is in the opposite direction. Thus, she will end up at a location \(\mathbf{\text{A}}+\left(\mathbf{\text{–B}}\right)\), or \(\mathbf{\text{A}}–\mathbf{\text{B}}\).

A vector labeled negative B is inclined at an angle of sixty-eight degrees below a horizontal line. A dotted line in the reverse direction inclined at one hundred and twelve degrees above the horizontal line is also shown.

We will perform vector addition to compare the location of the dock, \(\text{A }\text{+ }\mathbf{B}\), with the location at which the woman mistakenly arrives, \(\text{A + }\left(\text{–B}\right)\).

(1) To determine the location at which the woman arrives by accident, draw vectors \(\mathbf{\text{A}}\) and \(\mathbf{\text{–B}}\).

(2) Place the vectors head to tail.

(3) Draw the resultant vector \(\mathbf{R}\).

(4) Use a ruler and protractor to measure the magnitude and direction of \(\mathbf{R}\).

Vectors A and negative B are connected in head to tail method. Vector A is inclined with horizontal with positive slope and vector negative B with a negative slope. The resultant of these two vectors is shown as a vector R from tail of A to the head of negative B. The length of the resultant is twenty three point zero meters and has a negative slope of seven point five degrees.

In this case, \(R=\text{23}\text{.}\text{0 m}\) and \(\theta =7\text{.}\text{5º}\) south of east.

(5) To determine the location of the dock, we repeat this method to add vectors \(\mathbf{\text{A}}\) and \(\mathbf{\text{B}}\). We obtain the resultant vector \(\mathbf{\text{R}}\text{‘}\):

A vector A inclined at sixty six degrees with horizontal is shown. From the head of this vector another vector B is started. Vector B is inclined at one hundred and twelve degrees with the horizontal. Another vector labeled as R prime from the tail of vector A to the head of vector B is drawn. The length of this vector is fifty two point nine meters and its inclination with the horizontal is shown as ninety point one degrees. Vector R prime is equal to the sum of vectors A and B.

In this case \(R\text{ = 52.9 m}\) and \(\theta =\text{90.1º}\)  north of east.

We can see that the woman will end up a significant distance from the dock if she travels in the opposite direction for the second leg of the trip.

Because subtraction of a vector is the same as addition of a vector with the opposite direction, the graphical method of subtracting vectors works the same as for addition.

Multiplication of Vectors and Scalars

If we decided to walk three times as far on the first leg of the trip considered in the preceding example, then we would walk \(\text{3 }×\text{ 27}\text{.}\text{5 m}\), or 82.5 m, in a direction \(\text{66}\text{.}0\text{º}\) north of east. This is an example of multiplying a vector by a positive scalar . Notice that the magnitude changes, but the direction stays the same.

If the scalar is negative, then multiplying a vector by it changes the vector’s magnitude and gives the new vector the opposite direction. For example, if you multiply by –2, the magnitude doubles but the direction changes. We can summarize these rules in the following way: When vector \(\mathbf{A}\) is multiplied by a scalar \(c\),

  • the magnitude of the vector becomes the absolute value of \(c\)\(A\),
  • if \(c\) is positive, the direction of the vector does not change,
  • if \(c\) is negative, the direction is reversed.

In our case, \(c=3\) and \(A=27.5 m\). Vectors are multiplied by scalars in many situations. Note that division is the inverse of multiplication. For example, dividing by 2 is the same as multiplying by the value (1/2). The rules for multiplication of vectors by scalars are the same for division; simply treat the divisor as a scalar between 0 and 1.

Resolving a Vector into Components

In the examples above, we have been adding vectors to determine the resultant vector. In many cases, however, we will need to do the opposite. We will need to take a single vector and find what other vectors added together produce it. In most cases, this involves determining the perpendicular components of a single vector, for example the x – and y -components, or the north-south and east-west components.

For example, we may know that the total displacement of a person walking in a city is 10.3 blocks in a direction \(\text{29}\text{.0º}\) north of east and want to find out how many blocks east and north had to be walked. This method is called finding the components (or parts) of the displacement in the east and north directions, and it is the inverse of the process followed to find the total displacement. It is one example of finding the components of a vector. There are many applications in physics where this is a useful thing to do. We will see this soon in Projectile Motion , and much more when we cover forces in Dynamics: Newton’s Laws of Motion . Most of these involve finding components along perpendicular axes (such as north and east), so that right triangles are involved. The analytical techniques presented in Vector Addition and Subtraction: Analytical Methods are ideal for finding vector components.

Learn about position, velocity, and acceleration in the “Arena of Pain”. Use the green arrow to move the ball. Add more walls to the arena to make the game more difficult. Try to make a goal as fast as you can.

  • The graphical method of adding vectors \(\mathbf{A}\) and \(\mathbf{B}\) involves drawing vectors on a graph and adding them using the head-to-tail method. The resultant vector \(\mathbf{R}\) is defined such that \(\mathbf{\text{A}}+\mathbf{\text{B}}=\mathbf{\text{R}}\). The magnitude and direction of \(\mathbf{R}\) are then determined with a ruler and protractor, respectively.
  • The graphical method of subtracting vector \(\mathbf{B}\) from \(\mathbf{A}\) involves adding the opposite of vector \(\mathbf{B}\), which is defined as \(-\mathbf{B}\). In this case, \(\text{A}–\mathbf{\text{B}}=\mathbf{\text{A}}+\left(\text{–B}\right)=\text{R}\). Then, the head-to-tail method of addition is followed in the usual way to obtain the resultant vector \(\mathbf{R}\).
  • Addition of vectors is commutative such that \(\mathbf{\text{A}}+\mathbf{\text{B}}=\mathbf{\text{B}}+\mathbf{\text{A}}\) .
  • The head-to-tail method of adding vectors involves drawing the first vector on a graph and then placing the tail of each subsequent vector at the head of the previous vector. The resultant vector is then drawn from the tail of the first vector to the head of the final vector.
  • If a vector \(\mathbf{A}\) is multiplied by a scalar quantity \(c\), the magnitude of the product is given by \(\text{cA}\). If \(c\) is positive, the direction of the product points in the same direction as \(\mathbf{A}\); if \(c\) is negative, the direction of the product points in the opposite direction as \(\mathbf{A}\).

Conceptual Questions

Which of the following is a vector: a person’s height, the altitude on Mt. Everest, the age of the Earth, the boiling point of water, the cost of this book, the Earth’s population, the acceleration of gravity?

Give a specific example of a vector, stating its magnitude, units, and direction.

What do vectors and scalars have in common? How do they differ?

Two campers in a national park hike from their cabin to the same spot on a lake, each taking a different path, as illustrated below. The total distance traveled along Path 1 is 7.5 km, and that along Path 2 is 8.2 km. What is the final displacement of each camper?

At the southwest corner of the figure is a cabin and in the northeast corner is a lake. A vector S with a length five point zero kilometers connects the cabin to the lake at an angle of 40 degrees north of east. Two winding paths labeled Path 1 and Path 2 represent the routes travelled from the cabin to the lake.

If an airplane pilot is told to fly 123 km in a straight line to get from San Francisco to Sacramento, explain why he could end up anywhere on the circle shown in (Figure) . What other information would he need to get to Sacramento?

A map of northern California with a circle with a radius of one hundred twenty three kilometers centered on San Francisco. Sacramento lies on the circumference of this circle in a direction forty-five degrees north of east from San Francisco.

Suppose you take two steps \(\mathbf{\text{A}}\) and \(\mathbf{\text{B}}\) (that is, two nonzero displacements). Under what circumstances can you end up at your starting point? More generally, under what circumstances can two nonzero vectors add to give zero? Is the maximum distance you can end up from the starting point \(\mathbf{\text{A}}+\mathbf{\text{B}}\) the sum of the lengths of the two steps?

Explain why it is not possible to add a scalar to a vector.

If you take two steps of different sizes, can you end up at your starting point? More generally, can two vectors with different magnitudes ever add to zero? Can three or more?

Problems & Exercises

Use graphical methods to solve these problems. You may assume data taken from graphs is accurate to three digits.

Find the following for path A in (Figure) : (a) the total distance traveled, and (b) the magnitude and direction of the displacement from start to finish.

A map of city is shown. The houses are in form of square blocks of side one hundred and twenty meters each. The path of A extends to three blocks towards north and then one block towards east. It is asked to find out the total distance traveled the magnitude and the direction of the displacement from start to finish.

(a) \(\text{480 m}\)

(b) \(\text{379 m}\), \(\text{18.4º}\) east of north

Find the following for path B in (Figure) : (a) the total distance traveled, and (b) the magnitude and direction of the displacement from start to finish.

Find the north and east components of the displacement for the hikers shown in (Figure) .

north component 3.21 km, east component 3.83 km

Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements \(\mathbf{\text{A}}\) and \(\mathbf{\text{B}}\), as in (Figure) , then this problem asks you to find their sum \(\mathbf{\text{R}}=\mathbf{\text{A}}+\mathbf{\text{B}}\).)

In this figure coordinate axes are shown. Vector A from the origin towards the negative of x axis is shown. From the head of the vector A another vector B is drawn towards the positive direction of y axis. The resultant R of these two vectors is shown as a vector from the tail of vector A to the head of vector B. This vector R is inclined at an angle theta with the negative x axis.

Suppose you first walk 12.0 m in a direction \(\text{20º}\) west of north and then 20.0 m in a direction \(\text{40.0º}\) south of west. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements \(\mathbf{A}\) and \(\mathbf{B}\), as in (Figure) , then this problem finds their sum \(\text{R = A + B}\).)

In the given figure coordinates axes are shown. Vector A with tail at origin is inclined at an angle of twenty degrees with the positive direction of x axis. The magnitude of vector A is twelve meters. Another vector B is starts from the head of vector A and inclined at an angle of forty degrees with the horizontal. The resultant R of the vectors A and B is also drawn from the tail of vector A to the head of vector B. The inclination of vector R is theta with the horizontal.

\(\text{19}\text{.}\text{5 m}\), \(4\text{.}\text{65º}\) south of west

Repeat the problem above, but reverse the order of the two legs of the walk; show that you get the same final result. That is, you first walk leg \(\mathbf{B}\), which is 20.0 m in a direction exactly \(\text{40º}\) south of west, and then leg \(\mathbf{A}\), which is 12.0 m in a direction exactly \(\text{20º}\) west of north. (This problem shows that \(\mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A}\).)

(a) Repeat the problem two problems prior, but for the second leg you walk 20.0 m in a direction \(\text{40.0º}\) north of east (which is equivalent to subtracting \(\mathbf{\text{B}}\) from \(\mathbf{A}\) —that is, to finding \(\mathbf{\text{R}}\prime =\mathbf{\text{A}}-\mathbf{\text{B}}\)). (b) Repeat the problem two problems prior, but now you first walk 20.0 m in a direction \(\text{40.0º}\) south of west and then 12.0 m in a direction \(\text{20.0º}\) east of south (which is equivalent to subtracting \(\mathbf{\text{A}}\) from \(\mathbf{\text{B}}\) —that is, to finding \(\mathbf{\text{R}}\prime \prime =\mathbf{\text{B}}-\mathbf{\text{A}}=-\mathbf{\text{R}}\prime \)). Show that this is the case.

(a) \(\text{26}\text{.}\text{6 m}\), \(\text{65}\text{.}\text{1º}\) north of east

(b) \(\text{26}\text{.}\text{6 m}\), \(\text{65}\text{.}\text{1º}\) south of west

Show that the order of addition of three vectors does not affect their sum. Show this property by choosing any three vectors \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{C}\), all having different lengths and directions. Find the sum \(\text{A + B + C}\) then find their sum when added in a different order and show the result is the same. (There are five other orders in which \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{C}\) can be added; choose only one.)

Show that the sum of the vectors discussed in (Figure) gives the result shown in (Figure) .

\(\text{52}\text{.}\text{9 m}\), \(\text{90}\text{.}\text{1º}\) with respect to the x -axis.

Find the magnitudes of velocities \({v}_{\text{A}}\) and \({v}_{\text{B}}\) in (Figure)

On the graph velocity vector V sub A begins at the origin and is inclined to x axis at an angle of twenty two point five degrees. From the head of vector V sub A another vector V sub B begins. The resultant of the two vectors, labeled V sub tot, is inclined to vector V sub A at twenty six point five degrees and to the vector V sub B at twenty three point zero degrees. V sub tot has a magnitude of 6.72 meters per second.

Find the components of \({v}_{\text{tot}}\) along the x – and y -axes in (Figure) .

x -component 4.41 m/s

y -component 5.07 m/s

Find the components of \({v}_{\text{tot}}\) along a set of perpendicular axes rotated \(\text{30º}\) counterclockwise relative to those in (Figure) .

Intro to Physics for Non-Majors Copyright © 2012 by OSCRiceUniversity is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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3.2 Vector Addition and Subtraction: Graphical Methods

Vectors in two dimensions.

A vector is a quantity that has magnitude and direction. Displacement, velocity, acceleration, and force, for example, are all vectors. In one-dimensional, or straight-line, motion, the direction of a vector can be given simply by a plus or minus sign. In two dimensions (2-d), however, we specify the direction of a vector relative to some reference frame (i.e., coordinate system), using an arrow having length proportional to the vector’s magnitude and pointing in the direction of the vector.

Figure 3.9 shows such a graphical representation of a vector , using as an example the total displacement for the person walking in a city considered in Kinematics in Two Dimensions: An Introduction . We shall use the notation that a boldface symbol, such as D D size 12{D} {} , stands for a vector. Its magnitude is represented by the symbol in italics, D D size 12{D} {} , and its direction by θ θ size 12{θ} {} .

Vectors in this Text

In this text, we will represent a vector with a boldface variable. For example, we will represent the quantity force with the vector F F size 12{F} {} , which has both magnitude and direction. The magnitude of the vector will be represented by a variable in italics, such as F F size 12{F} {} , and the direction of the variable will be given by an angle θ θ size 12{θ} {} .

Vector Addition: Head-to-Tail Method

The head-to-tail method is a graphical way to add vectors, described in Figure 3.11 below and in the steps following. The tail of the vector is the starting point of the vector, and the head (or tip) of a vector is the final, pointed end of the arrow.

Step 1. Draw an arrow to represent the first vector (9 blocks to the east) using a ruler and protractor .

Step 2. Now draw an arrow to represent the second vector (5 blocks to the north). Place the tail of the second vector at the head of the first vector .

Step 3. If there are more than two vectors, continue this process for each vector to be added. Note that in our example, we have only two vectors, so we have finished placing arrows tip to tail .

Step 4. Draw an arrow from the tail of the first vector to the head of the last vector . This is the resultant , or the sum, of the other vectors.

Step 5. To get the magnitude of the resultant, measure its length with a ruler. (Note that in most calculations, we will use the Pythagorean theorem to determine this length.)

Step 6. To get the direction of the resultant, measure the angle it makes with the reference frame using a protractor. (Note that in most calculations, we will use trigonometric relationships to determine this angle.)

The graphical addition of vectors is limited in accuracy only by the precision with which the drawings can be made and the precision of the measuring tools. It is valid for any number of vectors.

Example 3.1

Adding vectors graphically using the head-to-tail method: a woman takes a walk.

Use the graphical technique for adding vectors to find the total displacement of a person who walks the following three paths (displacements) on a flat field. First, she walks 25.0 m in a direction 49.0º 49.0º size 12{"49" "." "0º"} {} north of east. Then, she walks 23.0 m heading 15.0º 15.0º size 12{"15" "." "º°"} {} north of east. Finally, she turns and walks 32.0 m in a direction 68.0° south of east.

Represent each displacement vector graphically with an arrow, labeling the first A A size 12{A} {} , the second B B size 12{B} {} , and the third C C size 12{C} {} , making the lengths proportional to the distance and the directions as specified relative to an east-west line. The head-to-tail method outlined above will give a way to determine the magnitude and direction of the resultant displacement, denoted R R size 12{R} {} .

(1) Draw the three displacement vectors.

(2) Place the vectors head to tail retaining both their initial magnitude and direction.

(3) Draw the resultant vector, R R size 12{R} {} .

(4) Use a ruler to measure the magnitude of R R size 12{R} {} , and a protractor to measure the direction of R R size 12{R} {} . While the direction of the vector can be specified in many ways, the easiest way is to measure the angle between the vector and the nearest horizontal or vertical axis. Since the resultant vector is south of the eastward pointing axis, we flip the protractor upside down and measure the angle between the eastward axis and the vector.

In this case, the total displacement R R size 12{R} {} is seen to have a magnitude of 50.0 m and to lie in a direction 7.0º 7.0º size 12{7 "." 0°} {} south of east. By using its magnitude and direction, this vector can be expressed as R = 50.0 m R = 50.0 m size 12{R" = 50" "." "0 m"} {} and θ = 7 . 0º θ = 7 . 0º size 12{θ=7 "." "0°"} {} south of east.

The head-to-tail graphical method of vector addition works for any number of vectors. It is also important to note that the resultant is independent of the order in which the vectors are added. Therefore, we could add the vectors in any order as illustrated in Figure 3.19 and we will still get the same solution.

Here, we see that when the same vectors are added in a different order, the result is the same. This characteristic is true in every case and is an important characteristic of vectors. Vector addition is commutative . Vectors can be added in any order.

(This is true for the addition of ordinary numbers as well—you get the same result whether you add 2 + 3 2 + 3 size 12{"2+3"} {} or 3 + 2 3 + 2 size 12{"3+2"} {} , for example).

Vector Subtraction

Vector subtraction is a straightforward extension of vector addition. To define subtraction (say we want to subtract B B size 12{B} {} from A A size 12{A} {} , written A – B A – B size 12{ "A" "-B"} {} , we must first define what we mean by subtraction. The negative of a vector B B is defined to be –B –B ; that is, graphically the negative of any vector has the same magnitude but the opposite direction , as shown in Figure 3.20 . In other words, B B size 12{B} {} has the same length as –B –B size 12{"-" "B"} {} , but points in the opposite direction. Essentially, we just flip the vector so it points in the opposite direction.

The subtraction of vector B B from vector A A is then simply defined to be the addition of –B –B to A A . Note that vector subtraction is the addition of a negative vector. The order of subtraction does not affect the results.

This is analogous to the subtraction of scalars (where, for example, 5 – 2 = 5 +  ( –2 ) 5 – 2 = 5 +  ( –2 ) size 12{"5 – 2 = 5 + " \( "–2" \) } {} ). Again, the result is independent of the order in which the subtraction is made. When vectors are subtracted graphically, the techniques outlined above are used, as the following example illustrates.

Example 3.2

Subtracting vectors graphically: a woman sailing a boat.

A woman sailing a boat at night is following directions to a dock. The instructions read to first sail 27.5 m in a direction 66.0º 66.0º size 12{"66" "." 0º} {} north of east from her current location, and then travel 30.0 m in a direction 112º 112º size 12{"112"º} {} north of east (or 22.0º 22.0º size 12{"22" "." 0º} {} west of north). If the woman makes a mistake and travels in the opposite direction for the second leg of the trip, where will she end up? Compare this location with the location of the dock.

We can represent the first leg of the trip with a vector A A , and the second leg of the trip with a vector B B size 12{B} {} . The dock is located at a location A + B A + B . If the woman mistakenly travels in the opposite direction for the second leg of the journey, she will travel a distance B B (30.0 m) in the direction 180º – 112º = 68º 180º – 112º = 68º south of east. We represent this as –B –B , as shown below. The vector –B –B has the same magnitude as B B but is in the opposite direction. Thus, she will end up at a location A + ( –B ) A + ( –B ) , or A – B A – B .

We will perform vector addition to compare the location of the dock, A  +  B A  +  B size 12{ ital "A ""+ "B} {} , with the location at which the woman mistakenly arrives, A +  ( –B ) A +  ( –B ) size 12{ bold "A + " \( bold "–B" \) } {} .

(1) To determine the location at which the woman arrives by accident, draw vectors A A size 12{A} {} and –B –B .

(2) Place the vectors head to tail.

(3) Draw the resultant vector R R size 12{R} {} .

(4) Use a ruler and protractor to measure the magnitude and direction of R R size 12{R} {} .

In this case, R = 23 . 0 m R = 23 . 0 m size 12{R"=23" "." "0 m"} {} and θ = 7 . 5º θ = 7 . 5º size 12{θ=7 "." "5° south of east"} {} south of east.

(5) To determine the location of the dock, we repeat this method to add vectors A A size 12{A} {} and B B size 12{B} {} . We obtain the resultant vector R ' R ' size 12{R'} {} :

In this case R  = 52.9 m R  = 52.9 m size 12{R" = 52" "." "9 m"} {} and θ = 90.1º θ = 90.1º size 12{θ="90" "." "1° north of east "} {}  north of east.

We can see that the woman will end up a significant distance from the dock if she travels in the opposite direction for the second leg of the trip.

Because subtraction of a vector is the same as addition of a vector with the opposite direction, the graphical method of subtracting vectors works the same as for addition.

Multiplication of Vectors and Scalars

If we decided to walk three times as far on the first leg of the trip considered in the preceding example, then we would walk 3  ×  27 . 5 m 3  ×  27 . 5 m size 12{"3 " times " 27" "." "5 m"} {} , or 82.5 m, in a direction 66 . 0 º 66 . 0 º size 12{"66" "." 0 { size 12{º} } } {} north of east. This is an example of multiplying a vector by a positive scalar . Notice that the magnitude changes, but the direction stays the same.

If the scalar is negative, then multiplying a vector by it changes the vector’s magnitude and gives the new vector the opposite direction. For example, if you multiply by –2, the magnitude doubles but the direction changes. We can summarize these rules in the following way: When vector A A size 12{A} {} is multiplied by a scalar c c size 12{c} {} ,

  • the magnitude of the vector becomes the absolute value of c c size 12{c} {} A A size 12{A} {} ,
  • if c c size 12{A} {} is positive, the direction of the vector does not change,
  • if c c size 12{A} {} is negative, the direction is reversed.

In our case, c = 3 c = 3 size 12{c=3} and A = 27.5 m A = 27.5 m size 12{"A= 27.5 m"} . Vectors are multiplied by scalars in many situations. Note that division is the inverse of multiplication. For example, dividing by 2 is the same as multiplying by the value (1/2). The rules for multiplication of vectors by scalars are the same for division; simply treat the divisor as a scalar between 0 and 1.

Resolving a Vector into Components

In the examples above, we have been adding vectors to determine the resultant vector. In many cases, however, we will need to do the opposite. We will need to take a single vector and find what other vectors added together produce it. In most cases, this involves determining the perpendicular components of a single vector, for example the x - and y -components, or the north-south and east-west components.

For example, we may know that the total displacement of a person walking in a city is 10.3 blocks in a direction 29 .0º 29 .0º size 12{"29" "." 0º} } {} north of east and want to find out how many blocks east and north had to be walked. This method is called finding the components (or parts) of the displacement in the east and north directions, and it is the inverse of the process followed to find the total displacement. It is one example of finding the components of a vector. There are many applications in physics where this is a useful thing to do. We will see this soon in Projectile Motion , and much more when we cover forces in Dynamics: Newton’s Laws of Motion . Most of these involve finding components along perpendicular axes (such as north and east), so that right triangles are involved. The analytical techniques presented in Vector Addition and Subtraction: Analytical Methods are ideal for finding vector components.

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AP®︎/College Physics 1

Course: ap®︎/college physics 1   >   unit 1, representations of motion.

  • Deriving displacement as a function of time, acceleration, and initial velocity
  • Plotting projectile displacement, acceleration, and velocity
  • Position vs. time graphs
  • Why distance is area under velocity-time line
  • (Choice A)   CD A CD
  • (Choice B)   BC B BC
  • (Choice C)   DE C DE
  • (Choice D)   AB D AB

Mathematical Representations in Physics Lessons

  • First Online: 03 July 2019

Cite this chapter

graphical representation physics definition

  • Marie-Annette Geyer 4 &
  • Wiebke Kuske-Janßen 4  

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Physics is characterized by the use of specific types of representations. These representations play an important role in the teaching and learning of physics. This chapter/article starts with a general description of representations from a cognitive sciences' and semiotics' view and presents the state of theory about representations in physics education with focus on mathematical ones. Based on this, an adapted classification of representations in physics lessons is presented. This is followed by theoretical considerations and empirical findings about the relevance of (different) representations for learning and understanding physics and students' difficulties in this context. Furthermore, two models are introduced that are part of current research and offer an approach to analyse different changes of representations in physics classes. Ultimately, several implications for teaching will be derived.

The authors contributed equally to this work.

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Geyer, MA., Kuske-Janßen, W. (2019). Mathematical Representations in Physics Lessons. In: Pospiech, G., Michelini, M., Eylon, BS. (eds) Mathematics in Physics Education. Springer, Cham. https://doi.org/10.1007/978-3-030-04627-9_4

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Speed and Velocity

Average velocity: a graphical interpretation.

Average velocity is defined as the change in position (or displacement) over the time of travel.

Learning Objectives

Contrast speed and velocity in physics

Key Takeaways

  • Average velocity can be calculated by determining the total displacement divided by the total time of travel.
  • The average velocity of an object does not tell us anything about what happens to it between the starting point and ending point.
  • Average velocity is different from average speed in that it considers the direction of travel and the overall change in position.
  • velocity : A vector quantity that denotes the rate of change of position with respect to time, or a speed with a directional component.

In everyday usage, the terms “speed” and “velocity” are used interchangeably. In physics, however, they are distinct quantities. Speed is a scalar quantity and has only magnitude. Velocity, on the other hand, is a vector quantity and so has both magnitude and direction. This distinction becomes more apparent when we calculate average speed and velocity.

Average speed is calculated as the distance traveled over the total time of travel. In contrast, average velocity is defined as the change in position (or displacement) over the total time of travel.

image

Average Velocity : The kinematic formula for calculating average velocity is the change in position over the time of travel.

The SI unit for velocity is meters per second, or m/s, but many other units (such as km/h, mph, and cm/s) are commonly used. Suppose, for example, an airplane passenger took five seconds to move -4 m (the negative sign indicates that displacement is toward the back of the plane ). His average velocity would be:

v = Δ x / t = -4m/5s = -0.8 m/s

The minus sign indicates that the average velocity is also toward the rear of the plane.

The average velocity of an object does not tell us anything about what happens to it between the starting point and ending point, however. For example, we cannot tell from average velocity whether the airplane passenger stops momentarily or backs up before he gets to the back of the plane. To get more details, we must consider smaller segments of the trip over smaller time intervals.

To illustrate the difference between average speed and average velocity, consider the following additional example. Imagine you are walking in a small rectangle. You walk three meters north, four meters east, three meters south, and another four meters west. The entire walk takes you 30 seconds. If you are calculating average speed, you would calculate the entire distance (3 + 4 + 3 + 4 = 14 meters) over the total time, 30 seconds. From this, you would get an average speed of 14/30 = 0.47 m/s. When calculating average velocity, however, you are looking at the displacement over time. Because you walked in a full rectangle and ended up exactly where you started, your displacement is 0 meters. Therefore, your average velocity, or displacement over time, would be 0 m/s.

image

Average Speed vs. Average Velocity : If you started walking from one corner and went all the way around the rectangle in 30 seconds, your average speed would be 0.47 m/s, but your average velocity would be 0 m/s.

Instantaneous Velocity: A Graphical Interpretation

Instantaneous velocity is the velocity of an object at a single point in time and space as calculated by the slope of the tangent line.

Differentiate instantaneous velocity from other ways of determining velocity

  • When velocity is constantly changing, we can estimate our velocity by looking at instantaneous velocity.
  • Instantaneous velocity is calculated by determining the slope of the line tangent to the curve at the point of interest.
  • Instantaneous velocity is similar to determining how many meters the object would travel in one second at a specific moment.
  • instantaneous : (As in velocity)—occurring, arising, or functioning without any delay; happening within an imperceptibly brief period of time.

Typically, motion is not with constant velocity nor speed. While driving in a car, for example, we continuously speed up and slow down. A graphical representation of our motion in terms of distance vs. time, therefore, would be more variable or “curvy” rather than a straight line, indicating motion with a constant velocity as shown below. (We limit our discussion to one dimensional motion. It should be straightforward to generalize to three dimensional cases.)

image

Motion with Changing Velocity : Motion is often observed with changing velocity. This would result in a curvy line when graphed with distance over time.

To calculate the speed of an object from a graph representing constant velocity, all that is needed is to find the slope of the line; this would indicate the change in distance over the change in time. However, changing velocity it is not as straightforward.

Since our velocity is constantly changing, we can estimate velocity in different ways. One way is to look at our instantaneous velocity, represented by one point on our curvy line of motion graphed with distance vs. time. In order to determine our velocity at any given moment, we must determine the slope at that point. To do this, we find a line that represents our velocity in that moment, shown graphically in. That line would be the line tangent to the curve at that point. If we extend this line, we can easily calculate the displacement of distance over time and determine our velocity at that given point. The velocity of an object at any given moment is the slope of the tangent line through the relevant point on its x vs. t graph.

image

Determining instantaneous velocity : The velocity at any given moment is defined as the slope of the tangent line through the relevant point on the graph

Instantaneous Velocity, Acceleration, Jerk, Slopes, Graphs vs. Time : This is how kinematics begins.

In calculus, finding the slope of curve f(x) at x=x 0 is equivalent to finding the first derivative:

[latex]\frac{\text{df}(\text{x})}{\text{dx}}|_{\text{x}=\text{x}_0}[/latex].

One interpretation of this definition is that the velocity shows how many meters the object would travel in one second if it continues moving at the same speed for at least one second.

  • Curation and Revision.  Provided by : Boundless.com.  License :  CC BY-SA: Attribution-ShareAlike
  • OpenStax College, College Physics. September 17, 2013.  Provided by : OpenStax CNX.  Located at :  http://cnx.org/content/m42096/latest/?collection=col11406/1.7 .  License :  CC BY: Attribution
  • velocity.  Provided by : Wiktionary.  Located at :  http://en.wiktionary.org/wiki/velocity .  License :  CC BY-SA: Attribution-ShareAlike
  • OpenStax College, College Physics. October 20, 2012.  Provided by : OpenStax CNX.  Located at :  http://cnx.org/content/m42096/latest/?collection=col11406/1.7 .  License :  CC BY: Attribution
  • Provided by : http://None.  Located at :  http://None .  License :  CC BY-SA: Attribution-ShareAlike
  • Provided by : Light and Matter.  Located at :  http://lightandmatter.com/lmb.pdf .  License :  CC BY: Attribution
  • instantaneous.  Provided by : Wiktionary.  Located at :  http://en.wiktionary.org/wiki/instantaneous .  License :  CC BY-SA: Attribution-ShareAlike
  • Instantaneous Velocity, Acceleration, Jerk, Slopes, Graphs vs. Time.  Located at :  http://www.youtube.com/watch?v=STcgrV2L4tw .  License :  Public Domain: No Known Copyright .  License Terms : Standard YouTube license

Speed and Velocity Copyright © by Tulsa Community College is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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14.2: Mathematical Description of a Wave

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In order to describe the motion of a wave through a medium, we can describe the motion of the individual particles of the medium as the wave passes through. Specifically, we describe the position of each particle using its displacement, \(D\) , from its equilibrium position. Consider our rope example in which a sine wave is propagating through a medium (the rope) in the positive \(x\) direction, as shown in Figure \(\PageIndex{1}\).

clipboard_ea88dc7355db992ec8a665a9a9c435d1f.png

The displacement, \(D\) , of each point at position, \(x\) , in the medium is shown on the vertical axis of Figure \(\PageIndex{2}\). The solid black line corresponds to a snapshot of the wave at time \(t=0\) . The wave has an amplitude, \(A=5\text{m}\) , a velocity, \(v=1\text{m/s}\) , and a wavelength, \(\lambda=4\text{m}\) . The dotted line corresponds to a snapshot of the wave one second later, at \(t=1\text{s}\) , when the wave has moved to the right by a distance \(vt=1\text{m}\) .

clipboard_ec36feffc772d3da27f3f78b0fdf9497e.png

It is important to note that Figure \(\PageIndex{2}\) is not restricted to describing transverse waves, even if the illustration suggests that the particles’ displacements (vertical axis) are perpendicular to the direction of propagation of the wave (horizontal). The quantity, \(D\) , that is plotted on the vertical axis corresponds to the displacement of a particle from its equilibrium position. That displacement could correspond to the longitudinal displacement of a particle in a longitudinal wave.

At time \(t=0\) (solid line), the displacement of each point in the medium, \(D(x, t=0)\) , as a function of their distance from the origin, \(x\) , can be described by a sine function:

\[D(x,t=0)=A\sin\left(\frac{2\pi}{\lambda}x \right) \]

This corresponds to the displacement being 0 at the origin and at any position, \(x\) , that is a multiple of the wavelength, \(\lambda\) .

If the wave moves with velocity \(v\) in the positive \(x\) direction, then at time \(t\) , the sine function in Figure \(\PageIndex{2}\) will have shifted to the right by an amount \(vt\) (dotted line). The displacement of a point located at position \(x\) at time \(t\) will be the same as the displacement of the point at position \(x-vt\) at time \(t=0\) . For example, in Figure \(\PageIndex{2}\) the displacement of the point \(x=2\text{m}\) at time \(t=1\text{s}\) is the same as the displacement of the point at position \(x-vt=1\text{m}\) at \(t=0\) .

We can state this condition as:

\[\begin{aligned} D(x,t) = D(x-vt, t=0)\end{aligned}\]

That is, at some time \(t\) , the displacement of a point at position \(x\) is found by finding the position of the point at \(x-vt\) at \(t=0\) . We already have an equation to find the displacement of a point at \(t=0\) . Using the above condition, we can modify Equation 14.2.1 to write a function for the displacement of a point at position \(x\) at time \(t\) :

\[\begin{aligned} D(x,t) = A\sin\left( \frac{2\pi}{\lambda}(x-vt) \right)\end{aligned}\]

Noting that \(v/\lambda= 1/T\) , we can write this as:

\[\begin{aligned} D(x,t) = A\sin\left( \frac{2\pi x}{\lambda}- \frac{2\pi t}{T} \right)\end{aligned}\]

In the above derivation, we assumed that at time \(t=0\) , the displacement at \(x=0\) was \(D(x=0, t=0)=0\) . In general, the displacement could have any value at \(x=0\) and \(t=0\) , so we can allow the wave to shift left or right by including a phase, \(\phi\) , which can be determined from the displacement at \(x=0\) and \(t=0\) :

\[ D(x,t) = A\sin\left(\frac{2\pi x}{\lambda}-\frac{2\pi t}{T}+\phi \right)\]

where \(\phi=0\) corresponds to the displacement being zero at \(x=0\) and \(t=0\) .

Exercise \(\PageIndex{1}\)

What is the value of the phase \(\phi\) if the displacement of the point at \(x=0\) is \(D=A/2\) at time \(t=0\) ?

  • \(\pi/6\) .
  • \(\pi/4\) .
  • \(\pi/3\) .
  • \(\pi/2\) .

The equation above is written in terms of the wavelength, \(\lambda\) , and period, \(T\) , of the wave. Often, one uses the “wave number”, \(k\) , and the “angular frequency”, \(\omega\) , to describe the wave. These are defined as:

\[k=\frac{2\pi}{\lambda}\]

\[\omega = \frac{2\pi}{T}\]

Using the wave number and the angular frequency removes the factors of \(2\pi\) in the expression for \(D(x,t)\) , which can now be written as:

\[D(x,t)=A\sin (kx-\omega t+\phi )\]

It is important to note that the wave number, \(k\) , has no relation to the spring constant that we used for springs.

Using Equation 14.1.1 , we can also relate the wave number and angular frequency to the speed of the wave:

\[\begin{aligned} v = \frac{\lambda}{T}=\frac{\frac{2\pi}{k}}{\frac{2\pi}{\omega}}=\frac{\omega}{k}\end{aligned}\]

The Wave Equation

In Chapter 13, we saw that any physical system whose position, \(x\) , satisfies the following equation:

\[\begin{aligned} \frac{d^2x}{dt^2}=-\omega^2 x\end{aligned}\]

will undergo simple harmonic motion with angular frequency \(\omega\) , and that \(x(t)\) can be modeled as:

\[\begin{aligned} x(t) = A\cos(\omega t + \phi)\end{aligned}\]

Similarly, any system, where the displacement of a particle as a function of position and time, \(D(x,t)\) , satisfies the following equation:

\[\frac{\partial ^{2}D}{\partial x^{2}}=\frac{1}{v^{2}}\frac{\partial ^{2}D}{\partial t^{2}}\]

is described by a wave that propagates with a speed \(v\) . The equation above is called the “one-dimensional wave equation” and would be obtained from modeling the dynamics of the system, just as the equation of motion for a simple harmonic oscillator can be obtained from Newton’s Second Law. For the harmonic oscillator, the properties of the system (e.g. mass and spring constant) determine the angular frequency, \(\omega\) . For a wave, the properties of the medium determine the speed of the wave, \(v\) .

We use partial derivatives in the wave equation instead of total derivatives because \(D(x,t)\) is multivariate. A possible solution to the one-dimensional wave equation is:

\[\begin{aligned} D(x,t) = A\sin\left( kx -\omega t + \phi \right)\end{aligned}\]

which is the function that we used in the previous section to describe a sine wave.

Furthermore, if multiple solutions to the wave equation, \(D_1(x,t)\) , \(D_2(x,t)\) , etc, exist, then any linear combination, \(D(x,t)\) , of the solutions will also be a solution to the wave equation:

\[\begin{aligned} D(x,t) = a_1D_1(x,t)+a_2D_2(x,t)+a_3D_3(x,t)+\dots\end{aligned}\]

This last property is called “the superposition principle”, and is the result of the wave equation being linear in \(D\) (it does not depend on \(D^2\) , for example). It is easy to check, for example, that if \(D_1(x,t)\) and \(D_2(x,t)\) satisfy the wave equation, so does their sum.

In three dimensions, the displacement of a particle in the medium depends on its three spatial coordinates, \(D(x,y,z,t)\) , and the wave equation in Cartesian coordinates is given by:

\[\begin{aligned} \frac{\partial ^{2}D}{\partial x^{2}}+\frac{\partial ^{2}D}{\partial y^{2}}+\frac{\partial ^{2}D}{\partial z^{2}}&=\frac{1}{v^2}\frac{\partial ^{2}D}{\partial t^{2}}\\[4pt]\end{aligned}\]

There are many functions that can satisfy this equation, and the best choice will depend on the physical system being modeled and the properties of the wave that one wishes to describe.

Graphical Representation of Data

Graphical representation of data is an attractive method of showcasing numerical data that help in analyzing and representing quantitative data visually. A graph is a kind of a chart where data are plotted as variables across the coordinate. It became easy to analyze the extent of change of one variable based on the change of other variables. Graphical representation of data is done through different mediums such as lines, plots, diagrams, etc. Let us learn more about this interesting concept of graphical representation of data, the different types, and solve a few examples.

Definition of Graphical Representation of Data

A graphical representation is a visual representation of data statistics-based results using graphs, plots, and charts. This kind of representation is more effective in understanding and comparing data than seen in a tabular form. Graphical representation helps to qualify, sort, and present data in a method that is simple to understand for a larger audience. Graphs enable in studying the cause and effect relationship between two variables through both time series and frequency distribution. The data that is obtained from different surveying is infused into a graphical representation by the use of some symbols, such as lines on a line graph, bars on a bar chart, or slices of a pie chart. This visual representation helps in clarity, comparison, and understanding of numerical data.

Representation of Data

The word data is from the Latin word Datum, which means something given. The numerical figures collected through a survey are called data and can be represented in two forms - tabular form and visual form through graphs. Once the data is collected through constant observations, it is arranged, summarized, and classified to finally represented in the form of a graph. There are two kinds of data - quantitative and qualitative. Quantitative data is more structured, continuous, and discrete with statistical data whereas qualitative is unstructured where the data cannot be analyzed.

Principles of Graphical Representation of Data

The principles of graphical representation are algebraic. In a graph, there are two lines known as Axis or Coordinate axis. These are the X-axis and Y-axis. The horizontal axis is the X-axis and the vertical axis is the Y-axis. They are perpendicular to each other and intersect at O or point of Origin. On the right side of the Origin, the Xaxis has a positive value and on the left side, it has a negative value. In the same way, the upper side of the Origin Y-axis has a positive value where the down one is with a negative value. When -axis and y-axis intersect each other at the origin it divides the plane into four parts which are called Quadrant I, Quadrant II, Quadrant III, Quadrant IV. This form of representation is seen in a frequency distribution that is represented in four methods, namely Histogram, Smoothed frequency graph, Pie diagram or Pie chart, Cumulative or ogive frequency graph, and Frequency Polygon.

Principle of Graphical Representation of Data

Advantages and Disadvantages of Graphical Representation of Data

Listed below are some advantages and disadvantages of using a graphical representation of data:

  • It improves the way of analyzing and learning as the graphical representation makes the data easy to understand.
  • It can be used in almost all fields from mathematics to physics to psychology and so on.
  • It is easy to understand for its visual impacts.
  • It shows the whole and huge data in an instance.
  • It is mainly used in statistics to determine the mean, median, and mode for different data

The main disadvantage of graphical representation of data is that it takes a lot of effort as well as resources to find the most appropriate data and then represent it graphically.

Rules of Graphical Representation of Data

While presenting data graphically, there are certain rules that need to be followed. They are listed below:

  • Suitable Title: The title of the graph should be appropriate that indicate the subject of the presentation.
  • Measurement Unit: The measurement unit in the graph should be mentioned.
  • Proper Scale: A proper scale needs to be chosen to represent the data accurately.
  • Index: For better understanding, index the appropriate colors, shades, lines, designs in the graphs.
  • Data Sources: Data should be included wherever it is necessary at the bottom of the graph.
  • Simple: The construction of a graph should be easily understood.
  • Neat: The graph should be visually neat in terms of size and font to read the data accurately.

Uses of Graphical Representation of Data

The main use of a graphical representation of data is understanding and identifying the trends and patterns of the data. It helps in analyzing large quantities, comparing two or more data, making predictions, and building a firm decision. The visual display of data also helps in avoiding confusion and overlapping of any information. Graphs like line graphs and bar graphs, display two or more data clearly for easy comparison. This is important in communicating our findings to others and our understanding and analysis of the data.

Types of Graphical Representation of Data

Data is represented in different types of graphs such as plots, pies, diagrams, etc. They are as follows,

Related Topics

Listed below are a few interesting topics that are related to the graphical representation of data, take a look.

  • x and y graph
  • Frequency Polygon
  • Cumulative Frequency

Examples on Graphical Representation of Data

Example 1 : A pie chart is divided into 3 parts with the angles measuring as 2x, 8x, and 10x respectively. Find the value of x in degrees.

We know, the sum of all angles in a pie chart would give 360º as result. ⇒ 2x + 8x + 10x = 360º ⇒ 20 x = 360º ⇒ x = 360º/20 ⇒ x = 18º Therefore, the value of x is 18º.

Example 2: Ben is trying to read the plot given below. His teacher has given him stem and leaf plot worksheets. Can you help him answer the questions? i) What is the mode of the plot? ii) What is the mean of the plot? iii) Find the range.

Solution: i) Mode is the number that appears often in the data. Leaf 4 occurs twice on the plot against stem 5.

Hence, mode = 54

ii) The sum of all data values is 12 + 14 + 21 + 25 + 28 + 32 + 34 + 36 + 50 + 53 + 54 + 54 + 62 + 65 + 67 + 83 + 88 + 89 + 91 = 958

To find the mean, we have to divide the sum by the total number of values.

Mean = Sum of all data values ÷ 19 = 958 ÷ 19 = 50.42

iii) Range = the highest value - the lowest value = 91 - 12 = 79

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Practice Questions on Graphical Representation of Data

Faqs on graphical representation of data, what is graphical representation.

Graphical representation is a form of visually displaying data through various methods like graphs, diagrams, charts, and plots. It helps in sorting, visualizing, and presenting data in a clear manner through different types of graphs. Statistics mainly use graphical representation to show data.

What are the Different Types of Graphical Representation?

The different types of graphical representation of data are:

  • Stem and leaf plot
  • Scatter diagrams
  • Frequency Distribution

Is the Graphical Representation of Numerical Data?

Yes, these graphical representations are numerical data that has been accumulated through various surveys and observations. The method of presenting these numerical data is called a chart. There are different kinds of charts such as a pie chart, bar graph, line graph, etc, that help in clearly showcasing the data.

What is the Use of Graphical Representation of Data?

Graphical representation of data is useful in clarifying, interpreting, and analyzing data plotting points and drawing line segments , surfaces, and other geometric forms or symbols.

What are the Ways to Represent Data?

Tables, charts, and graphs are all ways of representing data, and they can be used for two broad purposes. The first is to support the collection, organization, and analysis of data as part of the process of a scientific study.

What is the Objective of Graphical Representation of Data?

The main objective of representing data graphically is to display information visually that helps in understanding the information efficiently, clearly, and accurately. This is important to communicate the findings as well as analyze the data.

  • Math Article

Graphical Representation

Graphical Representation is a way of analysing numerical data. It exhibits the relation between data, ideas, information and concepts in a diagram. It is easy to understand and it is one of the most important learning strategies. It always depends on the type of information in a particular domain. There are different types of graphical representation. Some of them are as follows:

  • Line Graphs – Line graph or the linear graph is used to display the continuous data and it is useful for predicting future events over time.
  • Bar Graphs – Bar Graph is used to display the category of data and it compares the data using solid bars to represent the quantities.
  • Histograms – The graph that uses bars to represent the frequency of numerical data that are organised into intervals. Since all the intervals are equal and continuous, all the bars have the same width.
  • Line Plot – It shows the frequency of data on a given number line. ‘ x ‘ is placed above a number line each time when that data occurs again.
  • Frequency Table – The table shows the number of pieces of data that falls within the given interval.
  • Circle Graph – Also known as the pie chart that shows the relationships of the parts of the whole. The circle is considered with 100% and the categories occupied is represented with that specific percentage like 15%, 56%, etc.
  • Stem and Leaf Plot – In the stem and leaf plot, the data are organised from least value to the greatest value. The digits of the least place values from the leaves and the next place value digit forms the stems.
  • Box and Whisker Plot – The plot diagram summarises the data by dividing into four parts. Box and whisker show the range (spread) and the middle ( median) of the data.

Graphical Representation

General Rules for Graphical Representation of Data

There are certain rules to effectively present the information in the graphical representation. They are:

  • Suitable Title: Make sure that the appropriate title is given to the graph which indicates the subject of the presentation.
  • Measurement Unit: Mention the measurement unit in the graph.
  • Proper Scale: To represent the data in an accurate manner, choose a proper scale.
  • Index: Index the appropriate colours, shades, lines, design in the graphs for better understanding.
  • Data Sources: Include the source of information wherever it is necessary at the bottom of the graph.
  • Keep it Simple: Construct a graph in an easy way that everyone can understand.
  • Neat: Choose the correct size, fonts, colours etc in such a way that the graph should be a visual aid for the presentation of information.

Graphical Representation in Maths

In Mathematics, a graph is defined as a chart with statistical data, which are represented in the form of curves or lines drawn across the coordinate point plotted on its surface. It helps to study the relationship between two variables where it helps to measure the change in the variable amount with respect to another variable within a given interval of time. It helps to study the series distribution and frequency distribution for a given problem.  There are two types of graphs to visually depict the information. They are:

  • Time Series Graphs – Example: Line Graph
  • Frequency Distribution Graphs – Example: Frequency Polygon Graph

Principles of Graphical Representation

Algebraic principles are applied to all types of graphical representation of data. In graphs, it is represented using two lines called coordinate axes. The horizontal axis is denoted as the x-axis and the vertical axis is denoted as the y-axis. The point at which two lines intersect is called an origin ‘O’. Consider x-axis, the distance from the origin to the right side will take a positive value and the distance from the origin to the left side will take a negative value. Similarly, for the y-axis, the points above the origin will take a positive value, and the points below the origin will a negative value.

Principles of graphical representation

Generally, the frequency distribution is represented in four methods, namely

  • Smoothed frequency graph
  • Pie diagram
  • Cumulative or ogive frequency graph
  • Frequency Polygon

Merits of Using Graphs

Some of the merits of using graphs are as follows:

  • The graph is easily understood by everyone without any prior knowledge.
  • It saves time
  • It allows us to relate and compare the data for different time periods
  • It is used in statistics to determine the mean, median and mode for different data, as well as in the interpolation and the extrapolation of data.

Example for Frequency polygonGraph

Here are the steps to follow to find the frequency distribution of a frequency polygon and it is represented in a graphical way.

  • Obtain the frequency distribution and find the midpoints of each class interval.
  • Represent the midpoints along x-axis and frequencies along the y-axis.
  • Plot the points corresponding to the frequency at each midpoint.
  • Join these points, using lines in order.
  • To complete the polygon, join the point at each end immediately to the lower or higher class marks on the x-axis.

Draw the frequency polygon for the following data

Mark the class interval along x-axis and frequencies along the y-axis.

Let assume that class interval 0-10 with frequency zero and 90-100 with frequency zero.

Now calculate the midpoint of the class interval.

Using the midpoint and the frequency value from the above table, plot the points A (5, 0), B (15, 4), C (25, 6), D (35, 8), E (45, 10), F (55, 12), G (65, 14), H (75, 7), I (85, 5) and J (95, 0).

To obtain the frequency polygon ABCDEFGHIJ, draw the line segments AB, BC, CD, DE, EF, FG, GH, HI, IJ, and connect all the points.

graphical representation physics definition

Frequently Asked Questions

What are the different types of graphical representation.

Some of the various types of graphical representation include:

  • Line Graphs
  • Frequency Table
  • Circle Graph, etc.

Read More:  Types of Graphs

What are the Advantages of Graphical Method?

Some of the advantages of graphical representation are:

  • It makes data more easily understandable.
  • It saves time.
  • It makes the comparison of data more efficient.

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graphical representation physics definition

Very useful for understand the basic concepts in simple and easy way. Its very useful to all students whether they are school students or college sudents

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  1. Graphical Representations

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  8. 2.8 Graphical Analysis of One-Dimensional Motion

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