graphical representation physics definition

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

An Introduction to Motion

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

Importance of Graphs in Motion

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

Distance-Time Graph

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

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

Distance - Time Graph for Uniform Motion

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

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

Distance - Time Graph for Non-Uniform Motion

Displacement-Time Graph

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

Velocity-Time Graph

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

Velocity - Time Graph

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

Solved Examples

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

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

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

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

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

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

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

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

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

Time taken $t = 5\sec $

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

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

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

Interesting Facts

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

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

A speedometer is a good example of instantaneous speed.

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

FAQs on Graphical Representation of Motion

1. State the difference between speed and velocity.

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

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

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

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

2. Define positive acceleration and negative acceleration.

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

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

3. State the difference between distance and displacement.

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

• 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.

Suggested Videos

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

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

Browse more Topics under Motion

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

Distance Time Graph

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

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

Velocity and  Time Graph

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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Graphical Representation of Motion: Definition, Types, and Uses

Graphical Representation of Motion:  The concept of the graphical representation of motion provides an easy way to understand and analyse the motion of an object. Representing the data in a graph provides a simpler method to get the basic information about the various physical quantities involved in an event. We use line graphs to represent the motion of an object.

In the graphical representation of motion, the dependent quantity is taken along the \(Y\)-axis and the independent quantity is taken on the \(X\)-axis. Check out this article to learn more about the graphical representation of motion Class 9, its types, uses, and more.

What is Graphical Representation of Motion?

Graphical representation of motion uses graphs to represent an object’s motion. It is a convenient way to describe the nature of an object’s motion. This approach to expressing an object’s motion makes it simple to examine the change in the many physical quantities of an object in motion. Graphs are extremely useful in examining a body’s linear motion.

For example, the slope of a displacement-time graph indicates the velocity (and speed) with which the object is moving. It also gives an idea of whether the motion of the object is uniform or non-uniform.

Graphical Representation of Motion – Definition

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 three main types of graphical representation of motion are displacement versus time graph, velocity versus time graph, and acceleration versus time graph.

A. Displacement-Time Graph

In a displacement-time graph, the displacement is plotted on the \(Y\)-axis, and the time is plotted on the \(X\)-axis. The displacement-time graph represents the change in the position of an object with respect to time.

1. Displacement-time Graph for a Body at Rest

An object is said to be at rest when it does not change its position with respect to time. The displacement-time graph for a stationary body will be a straight line parallel to the time axis. The figure shown below represents the displacement-time graph for a body at rest.

2. Displacement-time Graph for a Body Moving with Uniform Velocity

When an object covers equal distances in equal intervals of time and travels in a straight line, the object travels with a uniform velocity. The displacement-time graph of a body moving with uniform velocity will be a straight line inclined to some angle with the time axis, as shown in the figure below.

3. Displacement-time Graph for a Body Moving with Non-uniform Velocity

When an object covers unequal distances in equal intervals of time in a particular direction or when it is changing its direction of motion, then it is said to be moving with a non-uniform velocity. In this case, the displacement-time graph will be a curved line. The displacement-time graph of a body moving with non-uniform velocity is shown in the figure below.

The slope of the displacement-time graph gives the velocity of the object. To calculate the velocity of the object, we use the following formula:

\(v = \frac{{\Delta s}}{{\Delta t}} = \frac{{{s_2} – {s_1}}}{{{t_2} – {t_1}}}\) Where \(v = \) velocity of the object, \(s = \) displacement covered by the object and \(t = \) time taken.

B. Velocity-Time Graph

The change in velocity of the body with time is represented by the velocity-time graph. In the velocity-time graph, the velocity of the body is represented along the \(y\)-axis, and the time is represented along the \(x\)-axis. The slope of the velocity-time graph gives the acceleration of the moving body. \(a = \frac{{\Delta v}}{{\Delta t}}\) Where, \(a = \) acceleration of the body, \(v = \) velocity of the body and \(t = \) time. The area enclosed under the velocity-time graph gives the displacement of the body.

1. Velocity-time Graph for a Body Moving with a Uniform Velocity

When a body moves with a uniform velocity, its magnitude and direction remain the same with time. The velocity-time graph for such a motion will be a straight line parallel to the time axis. In this case, the acceleration of the body is zero.

2. Velocity-time Graph for Bodies whose Velocities are Increasing and Decreasing with Time at a Constant Rate

The velocity-time graph for a body whose velocity is increasing at a uniform rate will be a straight line as shown in figure \(\left( f \right).\) The acceleration, in this case, will be positive. The velocity-time graph for a body whose velocity is decreasing at a uniform rate is shown in figure \(\left( g \right).\) The acceleration of the body will be negative if its velocity decreases with time.

3. Velocity-time Graph for a Body Moving with Variable Velocity

If a body is moving with a variable velocity, then the velocity-time graph for the body will be a curved line. In this case, the acceleration of the body is also changing with time.

C. Acceleration-Time Graph

The slope of the acceleration-time graph gives the quantity called the jerk. Jerk is the rate of change of acceleration. \({\text{Jerk}} = \frac{{\Delta a}}{{\Delta t}}\)

The area under the acceleration-time graph represents the change in velocity.

Uses of Graphical Representation of Motion

The uses of graphical representation of motion are as follows:

• The displacement-time graph indicates how the displacement of a body changes with time.
• The slope of the displacement-time graph can determine the velocity of the body.
• The slope of the velocity-time graph can determine the acceleration of the body.
• The area under the velocity-time graph determines the displacement of the body.

Solved Problems Based on Graphical Representation of Equation of Motion

Q.1. A car is moving on a straight road at a uniform acceleration. The velocity of the car at different times is given below. Draw its speed-time graph and calculate its acceleration and also find out the distance covered by it in \(30\) seconds.

Ans: The variation in the speed of the car moving on a straight road with time can be represented by the speed-time graph.

The slope of the speed-time graph gives the acceleration of the body. Therefore, the acceleration \(\left( a \right)\) of the car is given by the following formula: \(a = \frac{{{v_B} – {v_A}}}{{{t_B} – {t_A}}}\) Where \({{v_A} = }\) Speed of the car at point \(A.\) \( {v_B} = \)Speed of the car at point \(B\) \({t_A} = \)Time at which the car is having speed \({v_A}\) \({t_B} = \)Time at which the car is having speed \({v_B}\) By putting the values of the speed and time in the above formula, we get: \(a = \frac{{20 – 5}}{{30 – 0}} = 0.5\,{\text{m}}/{{\text{s}}^2}\) The area enclosed by the speed-time graph gives the distance covered by the body. Therefore, the distance \(\left( s \right)\) covered by the car in \(30\,{\text{s=}}\) area of \(OABC.\) \(s = \) area of rectangle \(OADC + \)area of triangle \(ABD\) \(s = OA \times OC + \frac{1}{2}\left({AD \times BD} \right)\) \(s = 5 \times 30 + \frac{1}{2}\left({30 \times 15} \right)\) \(s = 375\,{\text{m}}\)

Q.2. In the following figure, the position-time graph for an object at different times is shown. Calculate the speed of the object during \(C\) to \(D.\)

Ans: The change in the position of an object with time can be represented by the distance-time graph. In this graph, time is taken along the \(X\)–axis, and the distance is taken along the \(Y\)-axis. The slope of the distance-time graph gives the speed of the object. The speed of the object during \(C\) to \(D\) will be given by: \({v_{{\text{from}}\,\text{C}\,\text{to}\,\text{D}}} = \frac{{9 – 3}}{{9 – 7}} = \frac{6}{2} = 3\,{\text{cm/s}}\)

From this article, we can conclude that the graphical representation of motion can simplify the method of analysing and determining any quantity involved in the event of motion. The slope of the displacement-time graph represents the velocity of the object, and it also tells whether the object is moving with a constant velocity or with a variable velocity.

The slope of the velocity-time graph represents the acceleration of the object. The area enclosed by the velocity-time graph represents the displacement of the object.

FAQs on Graphical Representation of Motion

The most frequently asked questions on the graphical representation of the equation of motion are answered here:

Q.1: What are the two uses of graphical representation of motion? A: The uses of the graphical representation of motion are as follows: 1. The graphical representation of motion helps to interpret the information about the distance, speed, and acceleration of an object at any instant of time immediately upon looking. 2. The velocity of a moving object can be easily determined by the displacement-time graph.

Q.2: How many types of graphs are there in motion? A: There are two main types of graphs in motion, namely displacement-time graph and velocity-time graph.

Q.3: What is the graphical representation of linear motion? A: The representation of the motion of a body moving along a straight line pictorially by a graph is called the graphical representation of linear motion. For example, the displacement-time graph of a car moving along a straight road with a uniform velocity in a straight line that is not parallel to the time axis.

Q.4: How important is a graphical representation in describing the motion of an object? A: The graphical representation of the motion of an object not only makes the understanding of the nature of the move easier but also makes it simple to find the various parameters by looking at the graph itself. Some values of different parameters can be determined by observing the graph of the motion.

For example, if the graph of two objects which are moving with a constant velocity is plotted on the same distance-time graph, then the object for which the slope of the graph is maximum is moving with a greater speed.

Q.5: How do you find the velocity of an object from the graphical representation of motion? A: The velocity of an object is defined as the displacement per unit time. The velocity of an object can be determined by the slope or gradient of the displacement-time graph for an object. The steeper the slope faster will be the object. The formula to find the velocity\(\left( v \right)\) of the object from the displacement-time graph is given below:

\(v = \frac{ { {s_2} – {s_1}}}{ { {t_2} – {t_1}}}\) where, \({s_1}\) and \({s_2}\) are the displacements at time instant \({t_1}\) and \({t_2}\), respectively.

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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

• Derivation of Physics Formula
• Derivation Of Equation Of Motion

Derivation of Equations of Motion

Equations of motion of kinematics describe the basic concept of the motion of an object such as the position, velocity or acceleration of an object at various times. These three equations of motion govern the motion of an object in 1D, 2D and 3D. The derivation of the equations of motion is one of the most important topics in Physics. In this article, we will show you how to derive the first, second and third equation of motion by graphical method, algebraic method and calculus method.

Definition of Equations of Motion

Equations of motion, in physics, are defined as equations that describe the behaviour of a physical system in terms of its motion as a function of time.

There are three equations of motion that can be used to derive components such as displacement(s), velocity (initial and final), time(t) and acceleration(a). The following are the three equations of motion:

• First Equation of Motion : \(\begin{array}{l}v=u+at\end{array} \)
• Second Equation of Motion : \(\begin{array}{l}s=ut+\frac{1}{2}at^2\end{array} \)
• Third Equation of Motion : \(\begin{array}{l}v^2=u^2+2as\end{array} \)

To brush up on the basics of motion, refer to the article listed below: :

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Derivation of Equation of Motion

The equations of motion can be derived using the following methods:

• Derivation of equations of motion by Simple Algebraic Method
• Derivation of equations of Motion by Graphical Method
• Derivation of equations of Motion by Calculus Method

In the next few sections, the equations of motion are derived by all the three methods in a simple and easy to understand way.

Derivation of First Equation of Motion

For the derivation, let us consider a body moving in a straight line with uniform acceleration. Then, let the initial velocity be u , acceleration is denoted as a , the time period is denoted as t , velocity is denoted as v , and the distance travelled is denoted as s .

Derivation of First Equation of Motion by Algebraic Method

We know that the acceleration of the body is defined as the rate of change of velocity .

Mathematically, acceleration is represented as follows:

where v is the final velocity and u is the initial velocity.

Rearranging the above equation, we arrive at the first equation of motion as follows:

Derivation of First Equation of Motion by Graphical Method

The first equation of motion can be derived using a velocity-time graph for a moving object with an initial velocity of u , final velocity v , and acceleration a .

In the above graph ,

• The velocity of the body changes from A to B in time t at a uniform rate.
• BC is the final velocity and OC is the total time t .
• A perpendicular is drawn from B to OC , a parallel line is drawn from A to D , and another perpendicular is drawn from B to OE (represented by dotted lines).

The following details are obtained from the graph above:

The initial velocity of the body, u = OA

The final velocity of the body, v = BC

From the graph, we know that

BC = BD + DC

Therefore, v = BD + DC

v = BD + OA (since DC = OA )

v = BD + u (since OA = u ) ( Equation 1 )

Now, since the slope of a velocity-time graph is equal to acceleration a .

a = slope of line AB

Since AD = AC = t , the above equation becomes:

BD = at ( Equation 2 )

Now, combining Equation 1 & 2 , the following is obtained:

Derivation of First Equation of Motion by Calculus Method

Since acceleration is the rate of change of velocity, it can be mathematically written as:

Rearranging the above equation, we get

Integrating both the sides, we get

Rearranging, we get

Derivation of Second Equation of Motion

For the derivation of the second equation of motion, consider the same variables that were used for derivation of the first equation of motion.

Derivation of Second Equation of Motion by Algebraic Method

Velocity is defined as the rate of change of displacement. This is mathematically represented as:

If the velocity is not constant then in the above equation we can use average velocity in the place of velocity and rewrite the equation as follows:

Substituting the above equations with the notations used in the derivation of the first equation of motion, we get

From the first equation of motion, we know that v = u + at . Putting this value of v in the above equation, we get

On further simplification, the equation becomes:

Derivation of Second Equation of Motion by Graphical Method

From the graph above, we can say that

Distance travelled (s) = Area of figure OABC = Area of rectangle OADC + Area of triangle ABD

As OA=u and OC=t, the above equation becomes,

As BD =at (from the graphical derivation of 1st equation of motion), the equation becomes,

On further simplification, the equation becomes

Derivation of Second Equation of Motion by Calculus Method

Velocity is the rate of change of displacement.

Mathematically, this is expressed as

Rearranging the equation, we get

Substituting the first equation of motion in the above equation, we get

Integrating both sides, we get

On further simplification, the equations becomes:

Derivation of Third Equation of Motion

For the derivation of the third equation of motion, consider the same variables that were used for the derivation of the first and second equations of motion.

Derivation of Third Equation of Motion by Algebraic Method

Substituting the standard notations, the above equation becomes

From the first equation of motion, we know that

Rearranging the above formula, we get

Substituting the value of t in the displacement formula, we get

Derivation of Third Equation of Motion by Graphical Method

From the graph, we can say that

The total distance travelled, s is given by the Area of trapezium OABC .

s = ½ × (Sum of Parallel Sides) × Height

s = 1/2 x (OA + CB) x OC

Since, OA = u , CB = v , and OC = t

The above equation becomes

s = 1/2 x (u+v) x t

Now, since t = (v – u)/ a

The above equation can be written as:

s = ½ x ((u+v) × (v-u))/a

s = ½ x (v+u) × (v-u)/a

s = (v 2 -u 2 )/2a

Third equation of motion is obtained by solving the above equation:

Derivation of Third Equation of Motion by Calculus Method

We know that acceleration is the rate of change of velocity and can be represented as:

We also know that velocity is the rate of change of displacement and can be represented as:

Cross multiplying (1) and (2), we get

This is how we derive the three equations of motion by algebraic, graphical and calculus method.

Similar Reading:

Important derivations and formulae of Class 12 Physics

Top 10 NTSE Important Questions on Motion Class 9

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3.2 Vector Addition and Subtraction: Graphical Methods

Vectors in two dimensions.

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

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

Vectors in this Text

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

Vector Addition: Head-to-Tail Method

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

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

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

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

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

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

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

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

Example 3.1

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

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

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

(1) Draw the three displacement vectors.

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

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

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

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

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

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

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

Vector Subtraction

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

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

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

Example 3.2

Subtracting vectors graphically: a woman sailing a boat.

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

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

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

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

(2) Place the vectors head to tail.

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

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

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

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

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

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

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

Multiplication of Vectors and Scalars

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

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

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

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

Resolving a Vector into Components

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

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

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Graphical Representation of Motion: Types & Graphs

Muskan Shafi

Content Writer

Graphical Representation of Motion is an easy and effective way to understand and analyse the motion of an object. It helps to calculate and get the basic information about the various physical quantities during the motion more quickly by representing the data in a graph.

• Motion is defined as the change in the position of an object with respect to time. 
• The motion of an object can be represented easily on a graph to understand various related phenomena. 
• Graphs offer an easy way to understand the relationship between two physical quantities.
• Line graphs are used to depict the motion of an object.
• The dependent quantity is plotted on the Y-axis, while the independent quantity is plotted on the X-axis. 

Read More:   NCERT Solutions for Class 9 Science Motion

Key Terms: Graphical Representation of Motion, Distance Time Graph, Velocity, Velocity Time Graph, Motion, Acceleration-Time Graph

Graphical Representation of Motion

[Click Here for Sample Questions]

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.
• it makes analysing the change in numerous physical quantities for an object in motion simple.
• Graphs are extremely beneficial when studying the linear motion of an object.

The three main types of graphs covered under Graphical Representation of Motion are:

• Distance-Time Graph
• Velocity-Time Graph
• Acceleration-Time Graph

Read More: 

Distance is the overall distance travelled by an object.

• It is measured in units of ‘ meters ’.
• Distance is directly proportional to the time taken.
• A distance-time graph is used to study the distance travelled by an object in a specific amount of time.
• The Y-axis (left) represents distance, while the X-axis represents time (bottom).
• The slope of a distance-time graph gives us the speed of the object.
• Distance-time graphs can be made for a uniform speed or for a non-uniform speed. 

Distance-Time Graph for Uniform Motion

If an object is covering equal distances in equal time intervals, then it is said to have uniform motion . The distance-time graph for uniform motion is a straight line passing from the origin.

The distance-time graph for uniform motion is as follows:

Distance-Time Graph for Non-Uniform Motion

When an object covers unequal distances in equal intervals of time, it is said to be in non-uniform motion. The nature of such a graph shows a non-linear variation of the distance travelled by the object with time.

The distance-time graph for non-uniform motion is given as

Distance-Time Graph for Rest

When an object does not change its position with respect to time, it is said to be at rest. The distance-time graph for an object at rest will be a straight line parallel to the time axis as given below:

Read More:  ​ Motion Class 9 Important Questions

Velocity-Time Graph describes the variation in velocity with time for an object moving in a straight line. 

• Acceleration is the rate at which velocity varies with time.
• The slope of the velocity-time graph gives us the acceleration of an object.
• Velocity-Time Graph is a straight line in which the x-axis represents time, while the y-axis represents the velocity.
• They are popularly known as V-T graphs .

Acceleration = Change in Velocity / Time

Velocity-Time Graph for Uniform Velocity

In the case of uniform velocity, the magnitude and direction of the object remain the same with time. The velocity-time graph for uniform velocity will be a straight line parallel to the time axis as the acceleration of the body is zero.

Velocity-Time Graph for Acceleration 

Velocity-time graph for an object whose velocity is increasing at a uniform rate is a straight line given as follows: 

Velocity-Time Graph for Retardation

Retardation is defined as the negative acceleration which happens when the velocity decreases with time. Velocity-time graph for an object whose velocity is decreasing at a uniform rate is as follows:

Acceleration time graph is used to calculate the change in velocity during a specified time interval.

• The time taken by the object is on the x-axis, and the acceleration of the item is on the y-axis.
• The area under the graph represents the change in the velocity of the object over the given period of time.

Acceleration-Time Graph 

Check More: 

Things to Remember

• Graphical Representation of Motion is a method of representing a data set pictorially with the help of a line graph.
• A graph depicts the relationship between two data sets, where one physical quantity depends on the other physical quantity.
• The distance-time graph shows how a body's displacement changes over time.
• The slope of the distance-time graph is used to calculate the velocity of the body.
• The slope of the velocity-time graph is used to calculate the acceleration of the body.
• A positive slope on a velocity-time graph indicates object acceleration.
• A negative slope indicates object deceleration or retardation.

Sample Questions

Ques. What are the benefits of Graphical Representation of Motion? (2 Marks)

Ans. The following are some of the benefits of Graphical Representation of Motion:

• The graphical representation of motion helps in the interpretation of information about an object's distance, speed, and acceleration at any given time.
• The displacement-time graph can quickly determine the velocity of a moving item.

Ques. What does linear motion look like on the graph? (2 Marks)

Ans. The graphical depiction of linear motion is a pictorial representation of the motion of a body traveling along a straight line. The displacement-time graph of a car traveling along a straight road at a constant speed in a straight line that is not parallel to the time axis, for example.

Ques. What role does a graphical representation have in describing an object's motion? (2 Marks)

Ans. The graphical representation of an object's motion not only makes it easier to comprehend the nature of the movement but also makes it simple to locate the various parameters by simply looking at the graph. The graph of the motion can be used to determine some of the values of various parameters. For example, if the graphs of two objects moving at the same constant velocity are plotted on the same distance-time graph, the item with the greatest slope is moving at a faster rate.

Ques. What are the different types of graphs in motion? (2 Marks)

Ans. The three basic types of graphical representations of motion are:

Ques. What's the difference between a distance-time graph and one that shows displacement over time?  (2 Marks)

Ans. The most significant distinction between a distance-time graph and a displacement-time graph is that a distance-time graph always climbs and never decreases, whereas a displacement graph may go down.

Ques. What does an acceleration-time graph show?  (2 Marks)

Ans. The acceleration of an object traveling in a straight line is displayed against time in this graph. Acceleration measurements are plotted on the y-axis, while time values are plotted on the x-axis.

Ques. What is a velocity-time graph? (1 Mark)

Ans. A velocity-time graph depicts how a moving object's velocity changes over time. The slope of a velocity-time graph represents the moving object's acceleration.

Ques. How to find the velocity of an object from the graphical representation of motion?  (2 Marks)

Ques. Define uniform velocity and uniform acceleration.  (2 Marks)

Ans. An object   is said to move with uniform velocity if it covers equal displacement in equal intervals of time. Whereas, a body is said to move with uniform acceleration if equal changes in velocity take place in equal intervals of time.

Ques. Can there be zero displacement?  (2 Marks)

Ans.  Yes, displacement can be zero. The displacement of an object is the actual change in its position when it moves from one position to the other. Thus, if an object travels from point A to B and then returns back to point A again, then the total displacement is zero.

CBSE X Related Questions

1. one-half of a convex lens is covered with a black paper. will this lens produce a complete image of the object verify your answer experimentally. explain your observations., 2. balance the following chemical equations. (a) hno 3 +ca(oh) 2   \(→\)  ca(no 3 ) 2 + h 2 o  (b) naoh + h 2 so 4   \(→\)  na 2 so 4 + h 2 o  (c) nacl + agno 3   \(→\)  agcl + nano 3   (d) bacl + h 2 so 4   \(→\)  baso 4 + hcl, 3. draw the structure of a neuron and explain its function., 4. explain the following in terms of gain or loss of oxygen with two examples each.  (a) oxidation (b) reduction, 5. an electric bulb is rated 220 v and 100 w. when it is operated on 110 v, the power consumed will be –, 6. find the focal length of a convex mirror whose radius of curvature is 32 cm., similar science concepts, subscribe to our news letter.

Dot & Line Blog

Graphical Analysis of Motion Class 9

• Kanwal Hafeez
• July 13, 2023

Table of Contents

1. introduction to graphical analysis of motion:.

Motion is the change in position of an object over time. It is a fundamental concept in physics and understanding the various types of motion is crucial for comprehending the principles of physics. Graphical analysis of motion provides a visual representation of an object’s motion using graphs, which helps in analyzing and interpreting its behavior.

1.1 Motion and its Types:

Motion can be classified into different types based on the characteristics of an object’s movement. The main types of motion include:

• Translational Motion: This type of motion involves an object moving in a straight line, such as a car moving along a road or a ball rolling down a slope.
• Rotational Motion: Rotational motion occurs when an object rotates or spins around a fixed axis. Examples include the Earth rotating on its axis or a spinning top.
• Oscillatory Motion: Oscillatory motion involves repeated back-and-forth motion around a central position. Examples include a swinging pendulum or a vibrating guitar string.
• Periodic Motion : Periodic motion is a type of motion that repeats itself at regular intervals. Examples include the motion of a pendulum, the Earth’s revolution around the Sun, or the daily rise and fall of tides.

Supercharge your physics knowledge and excel in your studies with our online physics classes. Join our dynamic and interactive learning environment, where expert instructors will guide you through the fascinating world of physics. Ignite your passion for scientific discovery, unravel the mysteries of the universe, and unlock your full potential. Don’t miss out on this opportunity to gain a deep understanding of concepts, master problem-solving skills, and prepare for future academic pursuits. Enroll now and embark on a transformative journey of physics exploration. Take the leap towards success and join our online physics classes today.

1.2 Importance of Graphical Analysis in Understanding Motion:

Graphical analysis plays a crucial role in understanding motion as it provides a visual representation of an object’s position, velocity, and acceleration over time. By analyzing graphs, we can gather valuable information about an object’s motion that may not be immediately apparent from numerical data.

Graphs such as displacement-time, velocity-time, and distance-time graphs provide a concise and clear way to study the behavior of moving objects. They allow us to observe trends, identify patterns, and make predictions about the object’s future motion.

For example , a displacement-time graph can show us whether an object is moving at a constant speed or accelerating. A straight line on the graph indicates uniform motion, while a curved line indicates non-uniform motion. By calculating the slope of the line, we can determine the object’s velocity at different time intervals.

Similarly, a velocity-time graph helps us analyze an object’s acceleration. The slope of the graph represents the object’s acceleration, with a steeper slope indicating a higher acceleration. The area under the graph can also be used to calculate the object’s displacement.

By employing graphical analysis, we can interpret and compare motion data more effectively. It allows us to visualize the relationships between different variables and provides a deeper understanding of the fundamental principles of motion. Graphical analysis is a powerful tool for studying motion. It helps us interpret motion data, identify different types of motion, and make predictions about an object’s future behavior. By representing motion visually through graphs, we gain valuable insights into the physical world around us.

2. Understanding Displacement-Time Graphs:

Displacement-time graphs provide a visual representation of an object’s displacement as a function of time. They offer valuable insights into the motion of an object and can be analyzed to determine various characteristics of its movement.

2.1 Definition and Interpretation of Displacement-Time Graphs:

A displacement-time graph plots the displacement of an object on the y-axis against time on the x-axis. Displacement refers to the change in position of an object relative to its starting point.

On a displacement-time graph, the slope of the line connecting two points represents the object’s velocity. A steeper slope indicates a higher velocity, while a shallower slope corresponds to a lower velocity. If the line is horizontal, the object is at rest (zero velocity).

The area under the curve of a displacement-time graph represents the total displacement of the object. If the area is above the x-axis, it indicates a positive displacement, whereas a negative displacement is represented by an area below the x-axis.

2.2 Calculating Average Velocity from Displacement-Time Graphs:

The average velocity of an object can be calculated from a displacement-time graph by dividing the total displacement by the total time elapsed. 

Mathematically, average velocity (v_avg) can be expressed as: v_avg = Δd / Δt where Δd represents the change in displacement and Δt represents the change in time. For example, let’s consider a displacement-time graph where an object starts at a position of 2 meters and moves with a constant velocity of 3 meters per second for 5 seconds. The displacement-time graph would be a straight line with a slope of 3. The calculation for average velocity is as follows: v_avg = Δd / Δt v_avg = (final displacement – initial displacement) / (final time – initial time) v_avg = (5 – 2) meters / 5 seconds v_avg = 3/5 meters per second v_avg = 0.6 meters per second Therefore, the average velocity of the object over the given time interval is 0.6 meters per second.

2.3 Graphical Representation of Uniform Motion:

Uniform motion is characterized by a constant velocity, where an object covers equal distances in equal intervals of time. On a displacement-time graph, uniform motion is represented by a straight line with a constant slope.

For example, if an object moves with a velocity of 4 meters per second for 8 seconds, the displacement-time graph would be a straight line with a slope of 4. This indicates uniform motion.

2.4 Graphical Representation of Non-Uniform Motion:

Non-uniform motion refers to a situation where an object’s velocity changes over time. On a displacement-time graph, non-uniform motion is represented by a curved line, indicating varying velocities.

For instance, if an object starts from rest and gradually accelerates with time, the displacement-time graph would be a curved line with an increasing slope.

Displacement-time graphs provide valuable information about an object’s motion. The slope of the graph indicates velocity, the area under the curve represents displacement, and the shape of the line provides insights into the uniformity or non-uniformity of motion. By analyzing these graphs and using mathematical equations, we can derive various characteristics of an object’s movement.

3. Analyzing Velocity-Time Graphs:

Velocity-time graphs provide a graphical representation of an object’s velocity as a function of time. They offer valuable insights into the acceleration, uniformity of motion, and various characteristics of an object’s movement.

3.1 Understanding Velocity-Time Graphs:

A velocity-time graph plots the velocity of an object on the y-axis against time on the x-axis. Velocity represents the rate at which an object changes its position.

On a velocity-time graph, the slope of the line connecting two points represents the object’s acceleration. A positive slope indicates positive acceleration (object speeding up), while a negative slope represents negative acceleration (object slowing down). A horizontal line corresponds to constant velocity (zero acceleration).

3.2 Interpreting Different Sections of Velocity-Time Graphs:

Different sections of a velocity-time graph can provide insights into an object’s motion. For example:

• Constant Velocity: A straight horizontal line on the graph indicates constant velocity. In this case, the object maintains the same speed throughout the given time interval.
• Positive Acceleration : A positive slope on the graph indicates positive acceleration. It means the object’s velocity is increasing with time, and it is speeding up.
• Negative Acceleration: A negative slope on the graph represents negative acceleration. It indicates that the object’s velocity is decreasing with time, and it is slowing down.

3.3 Calculating Acceleration from Velocity-Time Graphs:

The acceleration of an object can be determined from a velocity-time graph by calculating the slope of the line. 

Mathematically, acceleration (a) can be expressed as: a = Δv / Δt where Δv represents the change in velocity and Δt represents the change in time. For example , suppose an object starts with an initial velocity of 10 m/s and uniformly accelerates to a final velocity of 30 m/s over a time interval of 5 seconds. The velocity-time graph would be a straight line with a positive slope of 4 m/s^2. The calculation for acceleration is as follows: a = Δv / Δt a = (final velocity – initial velocity) / (final time – initial time) a = (30 – 10) m/s / 5 s a = 20 m/s / 5 s a = 4 m/s^2 Therefore, the acceleration of the object over the given time interval is 4 m/s^2.

3.4 Graphical Representation of Uniform Acceleration:

Uniform acceleration occurs when an object’s velocity changes by an equal amount in equal intervals of time. On a velocity-time graph, uniform acceleration is represented by a straight line with a constant slope.

For instance, if an object accelerates from rest at a rate of 2 m/s^2, the velocity-time graph would be a straight line with a positive slope of 2.

Velocity-time graphs provide valuable information about an object’s motion. The slope of the graph indicates acceleration, and different sections of the graph reveal the object’s behavior, whether it is maintaining a constant velocity or experiencing positive or negative acceleration. By analyzing these graphs and using mathematical equations, we can derive various characteristics of an object’s movement.

4. Exploring Distance-Time Graphs:

Distance-time graphs provide a visual representation of an object’s distance traveled as a function of time. They are valuable tools for analyzing the motion of objects and understanding their speed and changes in speed over time.

4.1 Definition and Interpretation of Distance-Time Graphs:

A distance-time graph plots the distance traveled by an object on the y-axis against time on the x-axis. Distance represents the total path covered by an object.

On a distance-time graph, the slope of the line connecting two points represents the object’s speed. A steeper slope indicates a higher speed, while a shallower slope corresponds to a lower speed. If the line is horizontal, the object is at rest (zero speed).

The area under the curve of a distance-time graph represents the total distance covered by the object. If the area is above the x-axis, it indicates positive distance or displacement, whereas an area below the x-axis represents negative distance or displacement.

4.2 Calculating Average Speed from Distance-Time Graphs:

The average speed of an object can be calculated from a distance-time graph by dividing the total distance traveled by the total time elapsed. 

Mathematically, average speed (v_avg) can be expressed as: v_avg = total distance / total time For exampl e, suppose an object travels a total distance of 100 meters in a time interval of 10 seconds. The distance-time graph would show a straight line with a constant slope of 10. The calculation for average speed is as follows: v_avg = total distance / total time v_avg = 100 meters / 10 seconds v_avg = 10 meters per second Therefore, the average speed of the object over the given time interval is 10 meters per second

4.3 Graphical Representation of Constant Speed:

Constant speed refers to a situation where an object covers equal distances in equal intervals of time. On a distance-time graph, constant speed is represented by a straight line with a constant slope.

For instance, if an object travels at a constant speed of 20 meters per second for a duration of 5 seconds, the distance-time graph would be a straight line with a slope of 20.

4.4 Graphical Representation of Changing Speed:

Changing speed occurs when an object’s speed varies over time. On a distance-time graph, changing speed is represented by a curved line, indicating varying slopes.

For example, if an object starts from rest and gradually accelerates, the distance-time graph would show a curved line with an increasing slope.

Here is an example of a distance-time graph representing an object’s changing speed:

In the above graph, the distance (D) is plotted on the y-axis, and time (Time) is plotted on the x-axis. The graph starts at rest (zero speed) and gradually increases in slope, indicating an accelerating object. At time t1, the slope is relatively gentle, indicating a slower speed. As time progresses to t2, the slope becomes steeper, indicating an increased speed. Finally, at time t3, the slope is at its steepest, representing the highest speed attained by the object. This curved line on the distance-time graph illustrates the changing speed of the object. The increasing slope signifies an acceleration or a change in speed over time. Please note that the exact shape and slope of the curve will depend on the specific acceleration profile of the object. This example demonstrates the general concept of a changing speed represented by a curved line on a distance-time graph.

Distance-time graphs provide valuable information about an object’s motion. The slope of the graph indicates speed, the area under the curve represents distance or displacement, and the shape of the line provides insights into the object’s uniform or non-uniform speed. By analyzing these graphs and using mathematical equations, we can derive various characteristics of an object’s movement.

5. Applying Graphical Analysis to Real-World Scenarios:

5.1 analyzing motion of objects in everyday life:.

Graphical analysis of motion is a valuable tool for understanding and analyzing the motion of objects in everyday life. By examining distance-time, velocity-time, and displacement-time graphs, we can gain insights into the behavior and characteristics of various objects’ movements.

For example, let’s consider the motion of a car traveling along a straight road. By tracking its position over time, we can create a distance-time graph. The graph would show how the car’s distance from its starting point changes as time progresses. The slope of the line on the graph represents the car’s speed, with a steeper slope indicating higher speed.

Similarly, a velocity-time graph would provide information about the car’s acceleration. If the car is accelerating, the graph would show a positive slope, while a negative slope would indicate deceleration. A horizontal line would indicate that the car is maintaining a constant speed.

By analyzing these graphs, we can determine various aspects of the car’s motion, such as average speed, acceleration, and changes in speed over time.

To provide a visual representation, let’s consider an example of a distance-time graph for a car’s motion:

In this graph, the distance (D) is plotted on the y-axis, and time (Time) is plotted on the x-axis. The graph illustrates the changing distance covered by the car over time.

5.2 Solving Problems using Graphical Analysis of Motion:

Graphical analysis of motion can be applied to solve problems involving real-world scenarios. By using the information provided in graphs, we can determine unknown quantities and make predictions about future motion.

For example, consider a scenario where a cyclist is traveling along a straight road. We are given a velocity-time graph representing the cyclist’s motion. By analyzing the graph, we can determine the cyclist’s acceleration, average speed, and changes in speed over time.

Furthermore, we can solve problems such as finding the time it takes for the cyclist to reach a specific distance or determining the distance traveled by the cyclist within a given time interval. By applying mathematical equations and principles of graphical analysis, we can extract meaningful information from the graphs and use it to solve problems and gain insights into the motion of objects.

To illustrate, let’s consider an example of a velocity-time graph for a cyclist’s motion:

In this graph, the velocity (V) is plotted on the y-axis, and time (Time) is plotted on the x-axis. The graph depicts the changing velocity of the cyclist over time.

By analyzing this graph and using mathematical equations, we can determine various properties of the cyclist’s motion, such as acceleration, average speed, and changes in speed over time.

Graphical analysis of motion is a valuable tool for understanding and analyzing the motion of objects in real-world scenarios. By examining distance-time, velocity-time, and displacement-time graphs, we can gain insights into an object’s motion and solve problems related to its movement. Graphs provide visual representations of motion, allowing us to analyze patterns, make predictions, and derive meaningful information about the behavior and characteristics of objects in motion.

6. Review and Practice Exercises:

6.1 review of graphical analysis concepts:.

Before diving into practice problems, let’s review the key concepts of graphical analysis of motion. These concepts are essential for understanding and interpreting distance-time, velocity-time, and displacement-time graphs.

• Distance-Time Graphs: A distance-time graph represents the distance traveled by an object as a function of time. The slope of the line on the graph represents the object’s speed, and the area under the curve represents the total distance traveled.
• Velocity-Time Graphs: A velocity-time graph depicts an object’s velocity (speed and direction) as a function of time. The slope of the line on the graph represents the object’s acceleration, while the area under the curve represents the displacement.
• Displacement-Time Graphs: A displacement-time graph shows the displacement of an object as a function of time. The slope of the line represents the object’s velocity, and the area under the curve represents the total displacement.

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Understanding these concepts and their graphical representations is crucial for analyzing and interpreting the motion of objects.

6.2 Practice Problems for Graphical Analysis of Motion:

Now, let’s apply our knowledge of graphical analysis to solve practice problems. These problems will help reinforce the concepts and skills required to interpret and analyze motion graphs.

Problem 1: Distance-Time Graph

The distance-time graph below represents the motion of a cyclist. Analyze the graph and answer the following questions:

a) Calculate the average speed of the cyclist between t1 and t3. b) Determine the distance covered by the cyclist during the time interval from t1 to t2. c) Find the instantaneous speed of the cyclist at t2. Solution: a) To calculate the average speed, we need to find the total distance covered and the total time elapsed between t1 and t3. From the graph, we can see that the distance covered is 40 meters, and the time elapsed is 5 seconds. Average speed = total distance / total time Average speed = 40 meters / 5 seconds Average speed = 8 meters per second b) To determine the distance covered between t1 and t2, we need to find the difference in distance at those two time points. From the graph, we can see that the distance at t1 is 20 meters, and the distance at t2 is 30 meters. Distance covered = distance at t2 – distance at t1 Distance covered = 30 meters – 20 meters Distance covered = 10 meters c) The instantaneous speed at t2 can be determined by finding the slope of the line tangent to the curve at that point. In this case, we can estimate the instantaneous speed by observing the steepness of the line at t2. Let’s assume it to be approximately 6 meters per second.

Problem 2: Velocity-Time Graph

The velocity-time graph below represents the motion of a car. Analyze the graph and answer the following questions:

a) Calculate the average acceleration of the car between t1 and t3. b) Determine the displacement of the car during the time interval from t1 to t2. c) Find the instantaneous acceleration of the car at t2. Solution: a) To calculate the average acceleration, we need to find the change in velocity and the change in time between t1 and t3. From the graph, we can see that the change in velocity is 10 m/s, and the time elapsed is 5 seconds. Average acceleration = change in velocity / change in time Average acceleration = 10 m/s / 5 seconds Average acceleration = 2 m/s^2 b) The displacement of the car during the time interval from t1 to t2 can be determined by finding the area under the curve between those two time points. From the graph, we can see that the area is a triangle with a base of 4 seconds and a height of 10 m/s. Displacement = (1/2) * base * height Displacement = (1/2) * 4 seconds * 10 m/s Displacement = 20 meters c) The instantaneous acceleration at t2 can be determined by finding the slope of the line tangent to the curve at that point. In this case, we can estimate the instantaneous acceleration by observing the steepness of the line at t2. Let’s assume it to be approximately 3 m/s^2.

By practicing these types of problems, you can further enhance your understanding of graphical analysis and improve your ability to interpret and solve real-world motion scenarios.

Remember, when solving problems using graphical analysis, it’s crucial to carefully analyze the given graph, identify the relevant quantities, and apply the appropriate mathematical equations and principles to arrive at the correct answers.

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Taking online physics classes from Dot and Line Learning can offer several advantages for Class 9 students. Here are some reasons why you should consider taking their online physics classes:

• Comprehensive Curriculum: Dot and Line Learning provides a comprehensive curriculum that covers all the essential topics of Class 9 physics. They ensure that students receive a thorough understanding of the subject, including concepts, theories, and problem-solving techniques.
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• Interactive Learning Environment: Dot and Line Learning offers an interactive learning environment where students can actively engage with the material. Through multimedia presentations, virtual experiments, and interactive quizzes, students can enhance their understanding of physics concepts and develop critical thinking skills.
• Flexibility and Convenience: Online physics classes provide flexibility in terms of timing and location. Students can access the classes from the comfort of their homes and choose the schedule that suits them best. This flexibility allows students to manage their time effectively and balance their academic commitments.
• Personalized Attention : Dot and Line Learning strive to provide personalized attention to students. They offer opportunities for one-on-one interactions with instructors, enabling students to ask questions, seek clarifications, and receive individualized guidance. This personalized approach ensures that students receive the support they need to excel in physics.
• Supplementary Resources: In addition to live classes, Dot and Line Learning may provide supplementary resources such as study materials, practice worksheets, and recorded lectures. These resources can be beneficial for self-study and revision, allowing students to reinforce their understanding of physics concepts.
• Progress Tracking and Assessments: Dot and Line Learning may have systems in place to track students’ progress and conduct regular assessments. These assessments help students gauge their understanding of the subject, identify areas of improvement, and track their overall performance. This feedback-oriented approach ensures continuous learning and growth.
• Affordable and Cost-effective: Online physics classes from Dot and Line Learning may offer cost-effective options compared to traditional in-person classes. They may provide various package options that cater to different budgets, making quality education accessible to a wider range of students.

Overall, taking online physics classes from Dot and Line Learning can provide a convenient, interactive, and comprehensive learning experience for Class 9 students. It can help build a strong foundation in physics, develop problem-solving skills, and prepare students for future academic pursuits in the field of science and engineering.

graphical analysis of motion in Class 9 is essential for understanding and interpreting the behavior of objects in motion. By analyzing distance-time, velocity-time, and displacement-time graphs, students can gain valuable insights into various aspects of motion, such as speed, acceleration, and displacement. Graphical analysis provides a visual representation that helps students identify patterns, make predictions, and derive meaningful information about the motion of objects. Through the study of graphical analysis, students develop skills in interpreting and analyzing graphs, applying mathematical equations, and solving problems related to motion. This knowledge and understanding are foundational for further studies in physics and other related scientific disciplines.

Online physics classes from Dot and Line Learning offer numerous benefits for Class 9 students. Their comprehensive curriculum, expert instructors, interactive learning environment, and personalized attention contribute to a meaningful and effective learning experience. The flexibility and convenience of online classes allow students to manage their time efficiently and access the material from anywhere. Supplementary resources, progress tracking, and assessments further enhance students’ understanding and provide opportunities for self-assessment. Moreover, online classes can be cost-effective, making quality education accessible to a wider range of students. 

By choosing Dot and Line Learning for online physics classes , Class 9 students can build a strong foundation in physics, develop critical thinking skills, and be well-prepared for future academic endeavors in the field of science and engineering.

1. What is the graphical representation of motion in Class 9?

In Class 9, the graphical representation of motion involves plotting various parameters related to motion, such as distance, velocity, and acceleration, on graphs. These graphs help visualize and analyze the behavior of objects in motion over time. Common types of graphs used in graphical representation of motion include distance-time graphs, velocity-time graphs, and displacement-time graphs.

2. What is a graphical explanation of motion?

A graphical explanation of motion involves using graphs to visually represent and analyze the behavior of objects in motion. Graphs provide a concise and clear way to understand how different parameters, such as distance, velocity, and acceleration, change over time. By examining the shape, slope, and area under the curve of these graphs, we can derive information about the speed, direction, and changes in motion.

3. What are the different types of graphical analysis of motion?

There are several types of graphical analysis used to study motion, including:

• Distance-Time Graphs: These graphs plot the distance traveled by an object against time. They provide insights into the object’s speed and changes in speed.
• Velocity-Time Graphs : These graphs plot the velocity of an object against time. They help determine the object’s acceleration and changes in acceleration.
• Displacement-Time Graphs: These graphs plot the displacement of an object against time. They show the object’s velocity and changes in velocity.
• Acceleration-Time Graphs: These graphs plot the acceleration of an object against time. They provide information about changes in acceleration over time.

4. What is graphical analysis in physics?

Graphical analysis in physics refers to the use of graphs to analyze and interpret the behavior of objects in motion. It involves plotting relevant parameters, such as distance, velocity, acceleration, and displacement, against time to visually represent and study the motion of objects. By examining the shape, slope, and area under the curve of these graphs, physicists can derive information about the characteristics and changes in motion, such as speed, direction, and acceleration. Graphical analysis is a fundamental tool in physics for understanding and quantifying the principles and laws governing motion.

5. Why should I choose Dot and Line Learning for online physics classes?

There are several reasons why you should choose Dot and Line Learning for online physics classes. Firstly, they have experienced and qualified instructors who are experts in teaching physics. Their comprehensive curriculum covers all the essential topics of physics for Class 9.

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AP®︎/College Physics 1

Course: ap®︎/college physics 1   >   unit 1.

• Representations of motion
• Deriving displacement as a function of time, acceleration, and initial velocity
• Plotting projectile displacement, acceleration, and velocity
• Position vs. time graphs

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