## Graphical Representation of Data

Somebody else wrote this.

Graphs began to appear around 1770 and became common only around 1820. They appeared in three different places, probably independently. These three places were the statistical atlases of William Playfair, the indicator diagrams of James Watt, and the writings of Johann Heinrich Lambert. We should note as well the descriptive geometry of Gaspard Monge, which had an important indirect influence on the way that graphs developed. William Playfair's statistical graphs of the British economy were the best known of these early efforts. (See Figure 4.) He first presented them in his Commercial and Political Atlas of 1785. James Watt's indicator was another important early source of graphs, because it was one of the very first self-recording instruments. It drew a pressure-volume graph of the steam in the cylinder of an engine while it was in action. Recording instruments in the 19th century could not easily record numbers directly, and so they had to inscribe data by drawing a trace on paper or smoked glass. Thus recording instruments produced graphs by necessity, not by choice. Johann Heinrich Lambert was the only scientist in the eighteenth century to use graphs extensively. He drew many beautiful graphs in the 1760s and 1770s and used them not only to present data but also to average random errors by drawing the best curve through experimental data points. Lambert insisted that natural philosophy could be pursued successfully only by careful mathematical analysis of quantitative measurements taken with precision instruments. The natural arrangement for such measurements was a table of quantities relating the values. In his Pyrometrie Lambert gave tables showing the number of days in each month that the temperature reached a certain value. The numbers in these tables snaked back and forth in a most graphlike manner, and Lambert followed them up with actual graphs of temperature data. Thus by the 1790s graphs of several different forms were available for those who might want to use them, but for the most part they were ignored until the 1830s, when statistical and experimental graphs became much more common. (Hankins)

Another history lesson from The Canadian Museum of Making

Watt & Southern, c. 1796 The indicator was soon adapted to provide a written record of each individual application instead of merely a transient observation. This was a tremendous analytical breakthrough, allowing, as it did, an accurate picture to be formed of the pressure of steam at any time during the movement of the piston. The inspiration was due to John Southern (1758-1815), Watt's draughtsman, who recorded in a letter dated 14th March 1796 that he had 'contrived an instrument that shall tell accurately what power any engine exerts'.

Graphs of data serve the following purposes…

- to show what has happened
- to show the relationship between quantities
- to show distribution

There are then the following general types of graphs

- time series
- scatter plot
- histogram (a type of bar graph)

What about the axes?

- Independent Variable — usually plotted on the horizontal axis
- Dependent Variable — usually plotted on the vertical axis
- Explanatory Variable — usually plotted on the horizontal axis
- Response Variable — usually plotted on the vertical axis
- Categorical Variables — represented by different symbols on the same coordinate system
- Lurking Variables (Hidden Variables)

What's interesting?

- in calculus the result is called the derivative and the process is known as differentiation
- maximum and minimum
- inflection points
- in calculus the result is called the integral and the process is known as integration

## Motion Graphs: Explanation, Review, and Examples

- The Albert Team
- Last Updated On: March 31, 2022

When trying to explain how things move, physicists don’t just use equations – they also use graphs! Motion graphs allow scientists to learn a lot about an object’s motion with just a quick glance. This article will cover the basics for interpreting motion graphs including different types of graphs, how to read them, and how they relate to each other. Interpreting motion graphs, such as position vs time graphs and velocity vs time graphs, requires knowledge of how to find slope. If you need a review or find yourself having trouble, this article should be able to help.

What We Review

## Types of Motion Graphs

There are three types of motion graphs that you will come across in the average high school physics course – position vs time graphs, velocity vs time graphs, and acceleration vs time graphs. An example of each one can be seen below.

The position vs time graph (on the left) shows how far away something is relative to an observer.

The velocity vs time graph (in the middle) shows you how quickly something is moving, again relative to an observer.

Finally, the acceleration vs time graph (on the right) shows how quickly something is speeding up or slowing down, relative to an observer.

Because all of these are visual representations of a movement, it is important to know your frame of reference. We learned in our introduction to kinematics that two people can observe the same event but describe it differently depending upon where they stand. If this or anything about the position, velocity, and/or acceleration is still a bit confusing, revisit our kinematics post and our acceleration post before moving on.

## Describing Motion with Position vs Time Graph s

We typically start with position-time graphs when learning how to interpret motion graphs – generally because they’re the easiest to try to picture. Let’s look at the position vs time graph from above. We see that our vertical axis is Position (in meters) and that our horizontal axis is Time (in seconds). This means we know how far away an object has moved from our observer at any given time. This particular graph shows an object moving steadily away from our observer.

## Position vs Time Graph for Multi-Stage Motion

Let’s consider the graph and images below. We are still considering a position vs time graph, but this time we are looking at motion that changes. The car begins by moving 5 meters away from the observer in the first 5 seconds. After that, the car remains stopped 5 meters away from the observer for another 5 seconds. Finally, the car turns around and moves for 5 seconds back to its original position, 0 meters from the observer.

There are two key points that we can take from the example above. The first is that our position vs time graph shows how far away we are at any given time and nothing else. It cannot tell us distance or displacement – we would have to do a little mathematics to find those out. The second is that the change in position is not always positive. Here, we’ve defined moving to the right as positive. So, in the beginning, when the car was moving to the right, its position increased. In the end, when it moved back to the left, it was moving in the negative direction.

## Position vs Time Graph for Passing an Observer

This implies that the position could, potentially, go below the x-axis. Let’s look at an example combining these two points in practice.

This time, our car started to the right, and drove straight past our observer to the left. At t=0\text{ s} , the car was 10\text{ m} to the right of our observer, so its position was x=10\text{ m} . As it passed the observer, its position was x=0\text{ m} at t=5\text{ s} . The car then ended its journey 10\text{ m} to our observers left at t=10\text{ s} so that its final position was x=-10\text{ m} .

## Finding Distance and Displacement from a Position vs Time Graph

Example 1: constant position vs time graph.

We’ll continue working from the graph above as we have already pulled the important values from it. Because we have a simple, straight line we only need the values from the very beginning and very end of the car’s journey, which we already pulled out above:

- t_{1}=0\text{ s}
- x_{1}=10\text{ m}
- t_{2}=10\text{ s}
- x_{2}=-10\text{ m}

## Finding Distance From A Position-Time Graph

As we learned from our introduction to kinematics lesson, we know that the equation for distance is:

The problem here is that we didn’t pull out any d values from our position vs time graph, only x values. We can still use those, though. In general, if you take the absolute value of an x value, it can be thought of as a d value and plugged into our distance equation. So the d values we’ll be using are:

- d_{1}=\lvert x_{1} \rvert =\lvert 10\text{ m} \rvert = 10\text{ m}
- d_{2}=\lvert x_{2} \rvert =\lvert -10\text{ m} \rvert = 10\text{ m}

Now we can plug these values into our equation and solve for our distance.

## Finding Displacement From A Position vs Time Graph

Our equation for displacement is:

In this case, we will be using x_{2} as x_{f} and x_{1} as x_{i} as they represent the end and beginning of our movements, respectively. When we plug in our values, we find:

In this case, the negative sign makes sense as our line is moving down the graph and the car moved from right to left, which we had previously defined as positive to negative.

## Example 2: Changing Position vs Time Graph

Now that we know the basics of finding distance and displacement from a position vs time graph, let’s get a bit more in-depth. We’ll return to the graph about the car that moved forward, stopped, and then turned around and returned to its original position. The graph has been copied below for convenience.

## How to Find Distance From A Position vs Time Graph

Finding distance from these graphs can get a bit complex as you’ll need to find several different values. If you’ll notice, the slope of our graph changes regularly – the line seems to turn. Each segment with a unique slope requires our attention. So, we’ll need to look at t=0\text{ s} through t=5\text{ s} , t=5\text{ s} through t=10\text{ s} , and t=10\text{ s} through t=15\text{ s} .

We’ll want to look at the position value on the left and right of each side of those segments and find the absolute value of the delta between those values. These will serve as the d values that we will plug into our distance equation.

- d_{1}=\lvert 5\text{ m}-0\text{ m} \rvert =5\text{ m}
- d_{2}=\lvert 5\text{ m}-5\text{ m} \rvert =0\text{ m}
- d_{3}=\lvert 0\text{ m}-5\text{ m} \rvert =5\text{ m}

We can now plug all of these values into our equation and solve for distance.

## How to Find Displacement From A Position vs Time Graph

Finding displacement from a graph that changes how it’s moving is a bit easier than finding the distance. Because displacement only concerns the distance between the starting and ending positions of an object’s motion, we only need to find the position at the rightmost point on the graph ( t=15\text{ s} ) and the leftmost point on the graph ( t=0\text{ s} ). The positions at these times will serve as our x_{f} and x_{i} values respectively.

- x_{f}=0\text{ m}
- x_{i}=0\text{ m}

Now that we have these values, we can plug them into our displacement formula and solve:

## Finding Velocity from a Position vs Time Graph

Now that we know how to find distance and displacement from a position vs time graph, we can start finding another value – velocity. If you think about it, these distances and displacements that we’re finding are occurring over some amount of time (as given by the graph) and all we really need to find velocity is displacement and time. So let’s start with a simple graph – the one of an object moving steadily away.

The displacement for the movement depicted by this graph would be \Delta x=25\text{ m}-0\text{ m}=25\text{ m} and because our time here moves from t=0\text{ s} to t=5\text{ s} , we have a change in time of \Delta t=5\text{ s} . This is enough information for us to solve for the velocity using the equation we learned before:

One very important thing you may notice if you’re savvy with slopes is that the slope of this graph is also equal to 5 . (If you are not particularly savvy with slopes, I would recommend reviewing how to solve for slope as we’ll be relying on that knowledge for most of what remains of this post.) This similarity is no mere coincidence. The velocity of any movement will always be equal to the slope of the position-time graph at that time.

## Proving Velocity is the Slope of Position vs Time Graphs

The slope of any given straight line can be found with the equation

Here, m is the slope, y_{2} and y_{1} are two different position values, and x_{2} and x_{1} are the time values corresponding to the two position values.

Let’s begin by selecting two points off of our graph above (being sure to include the units when we do). Let’s take (2\text{ s},10\text{ m}) and (4\text{ s},20\text{ m}) and plug the values in:

This setup of subtracting the rightmost value from the leftmost value should look a bit familiar. Another way to think of it is as taking a final value and subtracting an initial value, much like a delta. In fact, this is equivalent to a change in position over a change in time – the definition of velocity.

Let’s make sure our slope works out to be a velocity value before we jump to any conclusions, though. If the slope of this graph is also the velocity of the same motion, two things need to be true. First, we need a numerical value of 5 . Second, we need our units to be in m/s. Let’s solve the equation above and see what we get.

We have a value that matches the velocity we solved for in both numerical value and physical units. We could have selected any two points on our graph and received the same result. The part of this that truly proves the slope of any position-time graph is the velocity is that the units of our slope work out to be m/s – the units for velocity.

## Example 1: Finding Velocity from a Position vs Time Graph

Let’s try finding our velocity again, but using the slope formula. We’ll reuse the graph below that we saw earlier in this article.

We’ll want to begin by selecting the points we want to use. This graph is a straight line which means its slope never changes so it won’t particularly matter what two points we choose. Since we have a point where our x value is zero and a separate point where our y value is zero, we may as well use those to make the mathematics easier. So, the values we’ll be using here are:

- y_{2}=0\text{ m}
- y_{1}=10\text{ m}
- x_{2}=5\text{ s}
- x_{1}=0\text{ s}

Now, all we need to do is set our velocity equal to our slope, plug in our values, and solve for our velocity:

Here, we get a negative velocity of v=-2\text{ m/s} . If we look at our graph, we see it has a negative slope, so we should have expected this negative velocity from the start. If you ever get a positive when you expected a negative or vice versa, check to make sure you plugged your values into your formula in the correct order. That simple mistake has thrown many scientists off course.

## Example 2: Finding Velocity with Changing Motion

Being able to find the velocity of a simple, straight position vs time graph is all well and good, but there will be times when you’ll have to split a graph apart. Let’s revisit the graph below as an example of this.

We already said before that we could split this graph up into a few different chunks based on when the slope changes. We know what happens when we have a positive slope and what happens when we have a negative slope, though, so let’s look at just the middle section where it’s flat. Here, the values we can pull from the line segment are:

- y_{2}=5\text{ m}
- y_{1}=5\text{ m}
- x_{1}=10\text{ s}

If we plug these values into our slope formula, we can find that

Since the segment was a flat line with a slope of 0 , the velocity also had to be 0\text{ m/s} . If we recall, this graph depicted a car that moved in the positive direction, stopped and remained motionless, and then moved back in the negative direction. The middle segment of this graph, the one that we looked at, corresponds to when the car was stopped so again. Therefore, it makes sense that we would see a velocity of 0\text{ m/s} .

## Describing Motion with Velocity vs Time Graph s

Velocity-time graphs are relatively similar to position-time graphs, and just as important in the study of motion graphs. We still have our time in seconds along the x-axis, but now we have our velocity in meters per second along the y-axis. Let’s consider the velocity-time graph below.

To find the velocity of an object at any given time here, we simply need to read the value from the graph. There’s no mathematics to do or formulas to use. So, for example, at t=2\text{ s} the velocity is v=4\text{ m/s} because that is the value we read off of the graph. Similarly, the velocity is t=4\text{ s} is v=8\text{ m/s} . The fact that those two values differ and that the slope here is positive tells us that the motion in this graph is an object moving away from an observer and getting faster – like a car leaving a stoplight. We can also show more complicated motions and dip below the x-axis.

## Velocity vs Time Graph with a Change in Direction

Let’s imagine a scenario for the graph to the right. We see that the graph starts with the object’s top velocity of v=10\text{ m/s} and then seems to get lower. The object reaches a velocity of v=0\text{ m/s} at t=2.5\text{ s} . While it may make sense to say that the object is now at the same point as the observer, we can’t actually infer that. All we can tell from here is that the object is momentarily at rest relative to the observer. The velocity then continues decreasing to v=-10\text{ m/s} , implying that the object is now moving in the negative direction. A real-life scenario for this may be that you observe someone pulling into a long driveway, stopping briefly at the end, and then backing down it.

Velocity vs Time Graph for Multi-Stage Motion

Now, let’s return to our car from before that moved in the positive direction, stopped, and then came back. Since the slope in each segment of the position graph was constant, we assumed that the car’s movements had a constant velocity and that it had zero velocity when it stopped. The velocity-time graph for this motion would look a bit like this:

You can see that the velocity remains a constant v=1\text{ m/s} while the car moves to the right, changes to v=0\text{ m/s} while the car stops, and then becomes v=-1\text{ m/s} while the car moves back to the left.

## Finding Displacement from a Velocity vs Time Graph

Much like how we could find a velocity from a position-time graph, we can find displacement from a velocity-time graph. This process will be a bit different. Instead of finding the slope of the velocity graph, we will be finding the area under the velocity graph. This may sound counterintuitive, but we can prove that it works by checking our units. Let’s say that an object moves at 5\text{ m/s} for 10\text{ s} . The velocity-time graph for this motion would look like this:

## Proving Displacement is the Area under Velocity vs Time Graphs

To prove that the area under this velocity-time graph is the object’s displacement, let’s start with figuring out the displacement. The equation for displacement is \Delta x=vt . In this case, we know v=5\text{ m/s} and t=10\text{ s} . Therefore, \Delta x=(5\text{ m/s})\cdot (10\text{ s})=50\text{ m} .

Now that we know we’re looking for a displacement of 50\text{ m} , let’s try finding the area under the curve. Specifically, this is the area between the line of our graph and the x-axis. We’ll start by drawing a shape – in this case, a rectangle. We’ll also include values for its base and height.

It’s worth noting here that the units along each axis were also included for the base and height of the rectangle. The equation for the area under the curve is the one you would use to find the area of a rectangle, A=bh . So, let’s pull down our values and solve our equation:

- b=10\text{ s}
- h=5\text{ m/s}

As a result, we obtained the same numerical value of 50 , but more to the point we obtained the correct units. The area under the curve of a velocity graph will always be a displacement. Let’s look at a couple of more examples. If you’re uncertain about your ability to remember the equations for the area of a rectangle or triangle, it may be worth writing them in your notes or referencing a formula sheet such as this one .

## Example 1: Finding Displacement for Multiple Velocities

The graph above was pretty simple, so let’s look at some more complex motion graphs. We can return to the velocity-time graph for our car that moved to the right, paused, then moved back to the left.

We already know that our displacement for this motion is 0\text{ m} . Let’s start by sectioning off our graph here into shapes we cam find the area of. Again, we’re looking for the area between the line of the graph and the x-axis.

It seems strange to have a negative value for the height of a shape as you’ve likely been told that area should always be a positive value. We’ll see why having a negative height when the graph is below the x-axis is both allowed and important. Now that we have all of our rectangles we can start finding their area.

Let’s begin with the rectangle farthest to the left.

- b=5\text{ s}
- h=1\text{ m/s}

Now we can solve for the area of our middle rectangle. This may seem like a trick question as it is, essentially, just a flat line, but we’ll still want to include it.

- h=0\text{ m/s}

Finally, let’s find the area of the rectangle on the right. This has a negative value for its height so it should also have a negative area, strange as that may seem.

- h=-1\text{ m/s}

Now that we know the area of all three rectangles, we’ll want to add those areas together to find the total area under the velocity-time curve and therefore also our total displacement.

Now, we see the expected value of 0\text{ m} that we’d found before. It’s important to note that this was only possible because one of our rectangles had a negative area.

## Example 2: Finding Displacement with Changing Velocity

We know we can use rectangles to find the area under a velocity-time graph, but not all graphs are horizontal lines. Sometimes, graphs are diagonal which requires us to find the area of a different shape – a triangle. Consider the velocity-time graph.

We can create a rectangle on the right where the velocity is constant, but the area where it’s increasing will not look like a rectangle at all. Instead, this is where we’ll have to create a section that is a triangle.

We now have two separate shapes. Much like when we had three separate rectangles, we’ll find the area of each shape individually and then add those two areas together to find the overall displacement for this motion. Let’s start with the triangle.

- h=10\text{ m/s}

Now, we can find the area of the rectangle portion.

Finally, we’ll add these two values to find our total displacement:

## Finding Acceleration from a Velocity vs Time Graph

At this point, it may not shock you to learn that the slope of a velocity-time graph can tell us just as much as the area under its curve. Instead of displacement, though, the slope of a velocity graph will tell us an object’s acceleration. Let’s consider the graph.

The velocity of the object being shown in this graph is steadily increasing by 2\text{ m/s} every 1\text{ s} . With that information, we can prove that a=\Delta v/\Delta t=(2\text{ m/s})/(1\text{ s})=2\text{ m/s}^2 . Now that we know what our acceleration should be, let’s try to find it by finding the slope of the velocity time graph.

## Proving Acceleration is the Slope of Velocity vs Time Graphs

If you remember from earlier, the slope of any given straight line can be found with the equation:

Let’s begin by selecting two points off of our graph above (being sure to include the units when we do). Let’s take (2\text{ s},4\text{ m/s}) and (4\text{ s},8\text{ m/s}) and plug the values in.

Again, this should look like a set of delta values – change in velocity over change in time, specifically. This is the definition of acceleration. While you may already be able to see how this will turn into proof that acceleration is the slope of a velocity graph, let’s keep going. What we’ll be looking for when we solve this equation this time will be a numerical value of 2 and units of m/s 2 .

Notice that we have a value that matches the acceleration we solved for before. We could have selected any two points on our graph and received the same result. The part of this that truly proves the slope of any velocity-time graph is the acceleration is that the units of our slope work out to be m/s 2 – the units for acceleration.

## Example 1: Finding Negative Acceleration

Let’s consider the velocity vs time graph.

We can see that the graph has a constant, negative slope so we can choose any two points we want and we should get the correct acceleration, which should also be a negative value. Whenever possible, it’s worth choosing values with zeros, so let’s select the points (0\text{ s},5\text{ m/s}) and (5\text{ s},0\text{ m/s}) . Now that we have our points, let’s pull out the values we need and plug them into our slope formula to solve for the acceleration of this object.

- y_{2}=0\text{ m/s}
- y_{1}=5\text{ m/s}

## Example 2: Finding Multiple Accelerations

Let’s consider a more complex example with the velocity-time graph below.

This graph has a change in its slope. This means we have two separate sections that we can look at: before t=5\text{ s} and after t=5\text{ s} .

Let’s consider the after t=5\text{ s} portion of the graph. Assuming that our acceleration will be negative because our velocity values are always negative is a common mistake among budding physicists. We’ll see here that this isn’t always true. Let’s choose the points (5\text{ s},-10\text{ m/s}) and (10\text{ s},-5\text{ m/s}) and plug these values into our slope formula.

- y_{2}=-5\text{ m/s}
- y_{1}=-10\text{ m/s}
- x_{2}=10\text{ s}
- x_{1}=5\text{ s}

If we plug these values into our slope formula, we can find that:

We see that even though our velocity values are negative, our slope is still positive so our acceleration must still be positive. Be careful when looking at motion graphs and making early assumptions. Things are sometimes more complicated than they appear.

## Describing Motion with Acceleration vs Time Graph s

Last but not least, we can describe an object’s motion with an acceleration vs time graph. These will likely be graphs with zero slope while you are starting your study of motion graphs. You may find them becoming more complicated if you pursue a career in physics, but for now, we can keep things simple. The below graph is a standard example of an acceleration graph you may see.

This graph actually shows acceleration due to gravity on Earth’s surface at a constant value of 9.81\text{ m/s}^2 . The other acceleration-time graph you’re likely to see in a high school physics class may look more like this:

This would indicate that the object’s velocity is not changing, or perhaps that it isn’t moving at all.

It is worth noting that the area under the curve of an acceleration vs time graph is equal to an object’s velocity, much like how the area under a velocity vs time graph is the displacement. Most high school physics classes won’t spend much time on this idea, but as you progress through your physics career this idea may come up. If you’d like to prove it to yourself, you could follow the same proof we used when proving the relationship between a velocity-time graph and an object’s displacement.

## Pairing Motion Graphs

The last skill we’ll cover for motion graphs is determining which pair of graphs represent the same motion. We can make more than one graph to describe any given motion. For example, we had both a position-time graph and a velocity-time graph for our car moving to the right, pausing, and then coming back to the left. We can extend this idea to include acceleration graphs too. Let’s consider the three graphs you were presented with at the beginning of this article.

We can see the position vs time graph on the left has a constant, positive slope. Since we know that velocity is the slope of a position vs time graph, our velocity must therefore be a constant, positive value. Indeed, we see that the velocity vs time graph here is constantly at 5\text{ m/s} . If we continue following this logic, we can assume that our acceleration should be zero as the slope of our velocity vs time graph is zero. And, again, we see that this holds true as the acceleration vs time graph is constant at 0\text{ m/s}^2 .

While it may be easy to see that these three motion graphs are connected after looking at them for a few moments, you’ll want to be able to compare more complex graphs throughout your physics career. There are a few steps you can take to achieve this goal.

## The Steps to Pairing Motion Graphs

Step 1: observe the shape and make a prediction.

Odds are, you’ll see three different shapes when looking at position graphs, three shapes when looking at velocity graphs, and only two shapes when looking at acceleration graphs. Without even looking at the numbers on these graphs, the shapes can tell you a lot. Here are the shapes of each graph you may see throughout your high school physics career:

All of these examples are of positive values but know that they may be flipped. This would simply mean that the slope values are negative instead of positive. Just from glancing at the graphs above, even though they don’t have any number, you could match them up just by looking at the slopes.

## Corresponding Shapes on Motion Graphs

Here’s a chart of how the different shapes match up.

You may notice that the arrows here go both ways and for good reason. While you’ll need to do some math to know exactly which parabola-shaped position-time graph matches a diagonal-shaped velocity-time graph, those two shapes will always go together, regardless of which you start with.

## Step 2: Decide if it is Positive or Negative

There is one more step you can take in matching up motion graphs before you start doing any actual math, which is looking for positive or negative slopes. Let’s look at this position-time graph.

We can see that the slope here is negative. The curve is above the x-axis so the values are positive, but the slope itself is negative. We know that this shape of the position-time graph will go with a flat velocity-time graph, but we need to pick the right one. We may need some mathematics for this, but let’s first try to narrow down our options. Let’s say you had to pick between the three velocity-time graphs below.

We’ve decided that our graph should be a straight, horizontal line. All three of these graphs match that description. We also said that our slope is negative, and only two of the velocity-time graphs have negative values. So, without doing any mathematics, we know that the positive-time graph above will have to be paired with the middle velocity-time graph or the right velocity-time graph. To find out for sure, we’ll have to add some numbers and do some math.

## Step 3: Calculate the Slope and Compare

Let’s take that same position-time graph and the two velocity-time graphs we couldn’t decide between. As you’ll see, they now have some numbers so that we can do some math to actually match the correct graphs.

We’ll need to begin by finding the slope of the position-time graph. To keep things simple, let’s use (0\text{ s},5\text{ m}) and (5\text{ s},0\text{ m}) . Now let’s pull out some values and solve for slope.

So, the velocity-time graph that matches our position-time graph here should have a value of -1\text{ m/s} . If we look, the velocity-time graph on the left has its line moving through -1\text{ m/s} so that must be the correct velocity-time graph.

We used these three steps of looking at the shape, looking at positive or negative, and then calculating the slope to go from a position-time graph to a velocity-time graph, but they can help us do much more than that. They can help us match up all three kinds of graphs or any pair of motion graphs – even a position-time graph and an acceleration-time graph.

## Example 1: Which Pair of Graphs Represent the Same Motion?

Let’s consider the position-time graph below and try to match it to the correct velocity vs time graph.

## Observe the Shape

Right off the bat, we know we have a parabola-shaped position-time graph. From that, we can narrow down our velocity-time graph options. A parabola position-time graph always goes to a diagonal velocity-time graph so we can cross off the middle graph immediately. Now we’re left with two different choices.

You may think you have the right answer already (and indeed you might), but let’s think through this carefully. We already matched up our shapes so now it’s time to compare our positives and negatives.

Decide if it is Positive or Negative

The parabola in the position-time graph points upward so it has a positive slope. That means our velocity-time graph needs to be positive. If you’re thinking too carefully about slope, you may be drawn to the graph on the left. While it’s true that the graph on the left has a positive slope, it actually contains negative values. The values are what’s important here, not the slope. Instead, the graph on the right is the correct choice here. We knew we needed a diagonal velocity-time graph with positive values and we only had one option. You may often encounter examples like this, but be careful to always check your instincts before answering too quickly.

It is also worth noting here that a helpful trick for recognizing whether a velocity-time graph could give a positive slope to a position-time graph is if the curve of the velocity-time graph is over the x-axis. The same is true in reverse; a velocity-time graph with a curve below the x-axis will match with a position-time graph with a negative slope.

## Example 2: Match the Velocity-Time Graph

The same principles that we just used above can also help us transition from a velocity-time graph to an acceleration-time graph. Let’s consider the set of motion graphs below.

From the start, we can see that we have a diagonal-shaped velocity-time graph so we can eliminate our middle acceleration-time graph. Although we will want a flat graph, the middle one is on the x-axis, which would imply that our velocity-time graph has zero slope. If that were the case, it would be flat instead of diagonal.

Now that we are again down to two graphs, let’s look at the positives and negatives. The velocity-time graph has a negative slope so we’ll want an acceleration-time graph with a curve below the x-axis. This leaves us with only one option – the graph on the right.

Again, we didn’t need to get to the mathematics. You could check the values if you wanted to, but often looking just at the shapes of your graphs will be enough. Just make sure you always think through both the shape of your motion graphs and if your positives and negatives line up.

Physicists use motion graphs to visualize data all the time. While the different types and shapes may be confusing at first, getting comfortable with them will help you make connections between the kinematics terms you’ve learned so far. It can also help you simplify problems by being able to visualize what the problem is asking you in a different way. If you take the time to get comfortable reading each time of motion graph, deriving different values from them, and matching them up, you’ll be well on your way to visualizing data the way research scientists do every day.

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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

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## 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

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

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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Mathematics in Physics Education pp 75–102 Cite as

## Mathematical Representations in Physics Lessons

- Marie-Annette Geyer 4 &
- Wiebke Kuske-Janßen 4
- First Online: 03 July 2019

1022 Accesses

9 Citations

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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Airey and Linder ( 2009 ) use the term semiotic resources and distinguish between tools, activities, and representations. In our argumentation this distinction is not important; tools and activities can be seen predominantly as handling with objective representations.

Fredlund et al. ( 2012 ) use the terminology of semiotic resources as well; therefore they call it the disciplinary affordance of a semiotic resource .

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Bossé, M. J., Adu-Gyamfi, K., & Cheetham, M. R. (2011). Assessing the difficulty of mathematical translations: Synthesizing the literature and novel findings. International Electronic Journal of Mathematics Education, 6 (3), 113–133.

Chandler, P. & Sweller, J. (1991). Cognitive load theory and the format of instruction. Cognition and Instruction , 8 (4), 293–332, as cited in Bauer (2015).

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Dolin, J. (2016). Representations in science. In R. Gunstone (Ed.), Encyclopedia of science education (pp. 836–838). Springer Science+Business Media Dordrecht 2015, Corrected Printing 2016.

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Eriksson, U., Linder, C., Airey, J., & Redfors, A. (2014). Introducing the anatomy of disciplinary discernment: an example from astronomy. European Journal of Science and Mathematics Education, 2 (3), 167–182.

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## Uniform Motion

- 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.

## 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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## 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 \(\PageIndex{1}\).

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 \(vt=1\text{m}\) .

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:

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

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.

## What is the pictorial or graphical representation?

Graphical representation refers to the use of charts and graphs to visually display, analyze, clarify, and interpret numerical data, functions, and other qualitative structures.

## What is graphical representation of motion class 9?

Graphical representation of motion: The most convenient way of studying the motion of a body is to construct a graph. In a graph: i) The magnitude of displacement, velocity or acceleration are plotted along the ordinates or the Y – axis ii) The time is plotted along the abscissa or the X – axis.

## What is the use of graphical representation of motion?

Graphical representation of motion is useful to study the nature of motion of bodies. In this type of representation, time is plotted along X-axis while distance or velocity is plotted along Y-axis. Graphs give more detailed information about the nature of motion than when motion is expressed in a tabular form.

## How many types of graphical representation of motion are there?

There are three basic types of graphical representations of motion. Displacement versus time graphs. Velocity versus time graphs. Acceleration versus time graphs.

## What is a pictorial example?

Pictorial is defined as something illustrated or expressed in pictures. If pictures tell a story of the history of a given plot of land, this is an example of a pictorial history. adjective.

## How do you write pictorial representation?

- Find the right icon. The power of pictorials is in their familiar shapes, so try to find icons that represent your categories.
- Avoid using very detailed icons.
- Avoid large data sets.

## What is a graph in physics?

There are so many things to learn about the purpose of the graph in Physics. A graph is the way of expressing the relationship between two quantities, out of which one alters as an after-effect from the other.

## What is motion class 9th physics?

Motion is a change in position of an object over time. Motion is described in terms of displacement, distance, velocity, acceleration, time and speed.

## How many types of graphs are there in physics class 9?

Displacement time graph, velocity-time graph, and acceleration time graph are three common types of graphs in classical mechanics.

## What are the uses of graph in physics class 9?

In describing motion graphs, that are used to illustrate object motion. It can be of numerous forms, like the graph between distance and time, which illustrates the fluctuation in distance with time, the graph between speed and time, the graph for projectile motion or Simple Harmonic Motion, and so on.

## What is the graphical representation of uniform speed?

The distance time graph of a body moving at uniform speed is always a straight line.

## What is graphical analysis of motion?

Graphical analysis is an approach to learning kinematics that uses slope and area relationships among motion graphs to solve for unknown variables.

## What are the advantages of graphical representation?

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.

## What is the importance of graphical representation of data?

Importance of Graphical Representation of Data It allows us to relate and compare data for different time periods. It is used in statistics to determine the mean, mode, and median of different data. It saves a lot of time as it covers most of the information in facts and figures. This in turn compacts the information.

## Is a pictorial representation of the entire system?

A chart is a pictorial representation of data.

## What’s a pictorial definition?

pic·to·ri·al pik-ˈtȯr-ē-əl. 1. : of or relating to a painter, a painting, or the painting or drawing of pictures. pictorial perspective. : of, relating to, or consisting of pictures.

## What is a pictorial representation called?

Pictorial representation of data is called pictograph.

## What is a pictorial symbol?

A pictogram, also called a pictogramme, pictograph, or simply picto, and in computer usage an icon, is a graphic symbol that conveys its meaning through its pictorial resemblance to a physical object.

## What is a pictorial representation of data using symbols?

A pictograph : Pictorial representation of data using symbols.

## What is visual representation?

Visual representation is mainly the direct or symbolic reflection of something in the format of photos, the images, memes, graphics to represent people, things, a place, or a situation.

## What are the 4 types of graphs in physics?

- 1 Distance-time Graphs.
- 2 Position-time Graphs or Displacement – Time Graphs.
- 3 Velocity-time Graphs.

## What is called graph?

Graph is a mathematical representation of a network and it describes the relationship between lines and points. A graph consists of some points and lines between them. The length of the lines and position of the points do not matter. Each object in a graph is called a node.

## How do you name a graph in physics?

- Your title should be short, but still clearly tell what you have graphed.
- The most common and recommended way to name your graph is to say what your “y” (vertical) and “x” (horizontal) axis are.
- As a rule of thumb, time almost always goes on the horizontal axis.

## What is difference between speed and velocity?

Speed is the time rate at which an object is moving along a path, while velocity is the rate and direction of an object’s movement. Put another way, speed is a scalar value, while velocity is a vector.

## What are the types of motion?

According to the nature of the movement, motion is classified into three types as follows: Linear Motion. Rotary Motion. Oscillatory Motion.

## Privacy Overview

## 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.

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.

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.)

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.

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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## 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.

## 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.

## 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.

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.

## 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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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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## Modulus of Elasticity

- Modulus Of Elasticity

## Modulus of Elasticity Definition

Modulus of Elasticity also referred to as Elastic Modulus or just Modulus is the quantification of the ratio of a material's elasticity. Modulus of elasticity measures the resistance of a material to non-permanent or elastic deformation when a ratio of stress is applied to its body. When under stress, materials will primarily expose their elastic properties. The stress induces them to deform, but the material will resume to its earlier state after the stress is eliminated. After undergoing the elastic region and through their production point, materials enter a plastic region, where they reveal everlasting deformation even after the tensile stress is removed.

## Graphical Representation of Modulus of Elasticity

Graphically, a Modulus is described as being the slope of the straight-line part of stress, denoted by (σ), and strain, denoted by (ε), curve. Focusing on the elastic region, if the slope is between two stress-strain points, the modulus will be the change in stress divided by the change in strain. Thus, Modulus =σ2−σ1 / ε2−ε1.

In this, the stress (σ) is force divided by the specimen's cross-sectional region and strain (ε) is the alteration in the length of the material divided by the material's original measure length. Seeing that both stress and strain are normalized quantifications, modulus exhibits a consistent material property that can be differentiated between specimens of different sizes. A huge steel specimen will have a similar modulus as a small steel specimen, though the large specimen would need a greater maximum force to deform the material.

Note: Brittle materials such as plastics, aluminum, copper and composites will reveal a steeper slope and higher modulus value than ductile materials such as iron, rubber, steel, etc.

Refer below for the Strain-Stress curve.

(Image will be uploaded soon)

## Modulus of Plastic or Elastomers

In contrast to brittle materials like metals and plastics, elastomeric materials do not display a yield point and continue to deform the material body elastically until they break. In the case of synthetic polymers having elastic properties like rubber, the modulus is simply expressed as a measure of the force at a given elongation. For example, in the graph below, the modulus is reported as stress at different levels for various materials.

## Types of Modulus of Elasticity

The modulus of elasticity of a material is the quantification of its stiffness and for most materials remains consistent over a range of stress. Following are the different types of modulus of elasticity:

## Young's Modulus

The ratio proportion of the longitudinal strain to the longitudinal stress is known as Young's modulus.

## Bulk Modulus

The ratio of the stress applied to the body on the body's fractional decrease in volume is called the bulk modulus. Thus, when a body is subjected to three mutually perpendicular stresses of the same intensity, the ratio of direct stress to the corresponding volumetric strain is what we call the Bulk Modulus. Bulk Modulus is generally denoted with the letter K.

## Shear Modulus

Also known as Rigidity Modulus, the ratio of the tangential force applied per unit area to the angular deformation in radians is called the shear modulus. Shear Modulus is generally denoted with the letter C.

## How to Calculate Different Types of Modulus of Elasticity

There are different types of modulus of elasticity and specific ways of calculating types of modulus of elasticity which we will be discussing below.

## Calculating Different Types of Modulus of Elasticity

Calculating the modulus of elasticity is generally required by users recording modulus. Hence, they should be well acquainted that there are various ways to measure the slope of the initial linear portion of a stress/strain curve. Remember that, when comparing outcomes of modulus for a given material between different laboratories, it is crucial to know which type of modulus calculation has been selected.

Thus, slopes are measured on the initial linear portion of the curve by employing least-squares fit on test data. The steepest slope is concluded as the modulus.

Let’s see below how to calculate different types of modulus of elasticity:

## Young’s Modulus

Young’s Modulus, usually denoted by (Y) = Longitudinal Stress ÷ Longitudinal Strain Nm-² or pascals.

Bulk Modulus of material is easily calculated in the following manner.

Bulk Modulus (K) = \[\frac {(F/A)} {(V/V)}\] = \[\frac {FV} {VA}\]

Thus, we get

K= \[\frac {PV} {v Nm^2}\]

Where, \[\frac {F} {A}\] = volume stress or bulk stress

\[\frac {v} {V}\] = volume strain or bulk strain

Thus, P = \[\frac {F} {A}\]

We determine the Shear Modulus in the following way.

Shear Modulus (n) = tangential stress ÷ Shearing strain

= \[\frac {F} {A Nm^2}\]

## Chord Modulus

To determine the chord modulus, we have to choose a beginning strain point and an end strain point. A line segment is needed to be drawn between the two points and the slope of that line is reported as the modulus.

Elastic modulus is identified using a standard linear regression strategy. The part of the curve to be used for the computation is chosen automatically and does not include the initial and final parts of the elastic deformation at the position where the stress-strain curve is non-linear.

## Hysteresis Modulus

Modulus is identified easily by a hysteresis loop produced by a portion of loading and reloading.

## Secant Modulus

Using the zero stress/strain point as the beginning value and a user-selected strain point as the end value, we can determine this type of modulus. A segment is constructed between the two points, and the slope of that line is reported as the modulus.

## Segment Modulus

We need to choose a start strain point and an end strain point. Using the least-squares fit on all points between the start and the endpoints, a line segment is drawn. The slope of the best fit line is thus recorded as the modulus.

## Tangent Modulus

Choosing a tangent point on the stress/strain curve, we can calculate the tangent type of modulus. The slope of the tangent line is thus recorded as the modulus.

## Does Young's Modulus Change with Change in Radius of the Wire?

No, Young's modulus does not change with a change in radius. Young's modulus is a material property, and for a wire its value remains constant. It doesn't depend on the dimension of material or the load(force) applied.

## FAQs on Modulus of Elasticity

1. What is the difference between Modulus of Elasticity and Modulus of Rigidity?

The main differences between Modulus of Elasticity and Modulus of Rigidity are:

Modulus of Elasticity describes the deformation of a material when a force acts at a right angle to the surface of the object whereas Modulus of Rigidity describes a material's deformation when a force is applied in a parallel direction to the surface of the object.

Change in Shape - When an object undergoes deformation force due to Modulus of Elasticity, it either gets lengthened or shortened. On the other hand, When an object undergoes some amount of force due to modulus of rigidity, it gets displaced with respect to another surface.

Relative Size - For most materials, the modulus of elasticity is more than the modulus of rigidity. But, the exception of this rule is the 'auxetic materials'. These materials have a negative value of Poisson’s ratios. Poisson's ratio is the change in the width of the material per unit width. The value of the Poisson's ratio is equal to the negative of the ratio of transverse strain to axial strain i.e, ( -transverse strain/axial strain).

2. Explain the relation between

(i) Modulus of Elasticity and Modulus of Rigidity

(ii) Modulus of Elasticity and bulk modulus

(i) The relation between Modulus of Elasticity and Modulus of Rigidity is given by:

E = 2G (1+μ)

Here, v refers to the number called the 'Poisson ratio' of the particular material.

Also, The fractional increase in length is defined as the axial strain and the fractional reduction in length is defined as the traverse strain on the object. The S.I unit of the relation between Young's modulus of Elasticity and Modulus of Rigidity is N/m2 or pascal(Pa).

(ii) The relation between Young’s modulus (modulus of elasticity) and bulk modulus is given by: E=3K(1−2μ).

Here, G = modulus of rigidity

.S.I unit between Young’s modulus and bulk modulus is N/m2 or pascal(Pa).

Also, the relation between Young’s modulus and bulk modulus k and modulus of rigidity is represented by :

E = \[\frac {9KG} {G+3K}\]

where, E = Young’s modulus or modulus of Elasticity.

3. Explain the following terms related to Young's Modulus:

(ii) Strain

Terms related to Young’s modulus or modulus of elasticity:

(i) Stress : Stress is the force experienced by the unit area of a material. Stress can be characterized on the basis of the direction of the deforming forces acting on the body. Magnitude of stress = F.A = Force.area

(ii) Strain : A force that stretches or affects the material and makes it reach a damaging degree. Strain = \[\frac {dL} {L}\]

, where dL = total elongation and L = original length.

4. What are the real-life examples of the Modulus of Elasticity?

Following are some practical and real-life examples of Young's modulus:

While walking up or down through the stairway, it is assumed that the young's modulus of the boards are good enough that they resist breaking, when we put our full body weight on them.

Springs are used in many mechanical and electronic devices in this modern world. Spring stores elastic energy which handles the load applied to it.

Aircraft and racing bicycles use young's modulus of elasticity. In these transport applications, stiffness is required at minimum weight.

When we apply a force on the rubber, then, there are major changes occurring in it. One of them is the 'stretching of rubber'. Due to Young's modulus of elasticity, rubber doesn't break but changes its length and handles the force.

The beams used in the bridge are an example of Young's modulus. They play an important role in reducing the stress and the bridge can withstand the huge load of the moving traffic.

5. Solve the following numerical on Young's Modulus of Elasticity :

A 45-kilogram traffic light is suspended from two steel cables of equal length and radii 0.50 cm. If each cable makes a 15° angle with the horizontal, what is the fractional increase in their length due to the weight of the light?

Given & Known Data:

m = 45 kg is the mass of the traffic light.

r = 0.50 cm = 0.0050 m is the radius of the steel cable.

0= 15° is the angle made by the cable with the horizontal.

E = 20 × 1010 N m 2 is Young's modulus of steel.

The traffic light is in static equilibrium condition. So in the vertical direction, we will be using the translational equilibrium condition.

sum Fy =0

T x sin + T x sinθ - W = 0

T x sinθ + T x sinθ = W

T x sinθ + T x sinθ = mg

T - \[\frac {(mg)} {(2\sin\theta)}\]

= 45 kg x 9.81 m s 2 2 x sin 15

= approx 852.82 N

Therefore, the tension in the cable is 852.82 N.

Now, we'll calculate the fractional increase in the length of the cable :

Delta x L = \[\frac {(TL)} {(AE)}\]

(Delta x L)/L = \[\frac {T} {(AE)}\]

= T ( pi r 2 )E

= 5.4 x 10 5

Hence, Due to the weight of traffic lights, the fractional increase in length of the cable is 5.4 x 10 5 .

## IMAGES

## VIDEO

## COMMENTS

Activities How do you explain kinematics? Kinematics is the study of motion without considering its causes. It includes the object's position, how fast it moves, and how its movement varies per...

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.

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.)

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

The graphical representation of motion is a method of representing a set of variables (physical quantities) pictorially with the help of a line graph where one physical quantity depends on the other physical quantity. Types of Graphical Representation of Motion

The inspiration was due to John Southern (1758-1815), Watt's draughtsman, who recorded in a letter dated 14th March 1796 that he had 'contrived an instrument that shall tell accurately what power any engine exerts'. Graphs of data serve the following purposes…. to show what has happened.

There are three types of motion graphs that you will come across in the average high school physics course - position vs time graphs, velocity vs time graphs, and acceleration vs time graphs. An example of each one can be seen below. The position vs time graph (on the left) shows how far away something is relative to an observer.

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.

Sample Problems and Solutions. Kinematic Equations and Graphs. Kinematics is the science of describing the motion of objects. Such descriptions can rely upon words, diagrams, graphics, numerical data, and mathematical equations. This chapter of The Physics Classroom Tutorial explores each of these representations of motion using informative ...

This definition needs specification for our purpose of describing mathematical representations in physics lessons. First of all the definition differentiates between the external world and our imagination. ... The form of a representation (e.g., how to read a graph) The relation between the representation and the domain (e.g., what the symbols ...

2.1 Definition of graphical representation design A graphical representation (or "diagram") is the product of making abstraction of some of the real-world complexity (Nelson et al., 2012; Rockwell & Bajaj, 2005) by purposefully representing selected information objects and their relationships, in the context of a specific

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.

Graphical Representation of Motion is the representation of the motion of an object using line graphs. It is a method of representing a set of variables graphically with the help of a line graph where one physical quantity depends on the other physical quantity. It is a convenient and effective way to describe the nature of the motion of an object.

The displacement-time graph below represents the motion of a toy car moving along a specially designed track. ... Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone ...

Answer. The equation above is written in terms of the wavelength, λ λ, and period, T T, of the wave. Often, one uses the "wave number", k k, and the "angular frequency", ω ω, to describe the wave. These are defined as: k = 2π λ (14.2.3) (14.2.3) k = 2 π λ. ω = 2π T (14.2.4) (14.2.4) ω = 2 π T.

A common graphical representation of motion along a straight line is the v vs. t graph, that is, the graph of (instantaneous) velocity as a function of time. In this graph, time is plotted on the horizontal axis and velocity on the vertical axis. Note that by definition, velocity and acceleration are vector quantities.

This review covers the definition of a vector, graphical and algebraic representations, adding vectors, scalar multiples, dot product, and cross product for two and three dimensional vectors, along with some physics applications. They are followed by several practice problems for you to try, covering all the basic concepts covered in the

What is graphical representation of motion class 9? Graphical representation of motion: The most convenient way of studying the motion of a body is to construct a graph. In a graph: i) The magnitude of displacement, velocity or acceleration are plotted along the ordinates or the Y - axis ii) The time is plotted along the abscissa or the X ...

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 ...

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.

Ohm's Law | Definition | Explanation | Graphical representation | MDCAT PhysicsOhm's Law | Definition | Explanation | Graphical representationOhm's Law | Def...

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.

Graphical Representation of Modulus of Elasticity. Graphically, a Modulus is described as being the slope of the straight-line part of stress, denoted by (σ), and strain, denoted by (ε), curve. Focusing on the elastic region, if the slope is between two stress-strain points, the modulus will be the change in stress divided by the change in ...