Position-Time Graphs & Velocity - Online Tutor, Practice Problems & Exam Prep

Position-time graphs illustrate an object's position over time, with the y-axis representing position and the x-axis representing time. The slope of the graph indicates velocity, where upward slopes signify forward motion, horizontal lines indicate a stop, and downward slopes represent backward movement. Curved graphs indicate changing velocity and non-zero acceleration. Average velocity is calculated between two points using the slope formula Δx × Δt, while instantaneous velocity is determined using a tangent line at a specific point.

Intro to Position-Time Graphs & Velocity

Hey, guys. So sometimes in your problems, you're going to be given graphs instead of numbers. In this video, we're going to talk about position-time graphs and how we use them to calculate things like velocity. Let's check it out. Now, guys, the basic idea is that a position-time graph just shows you an object's position on the y-axis, so right here, versus time on the x-axis over here. That might be kind of confusing because we're going to use position which is given by the variable x, but it's actually on the y-axis or the vertical axis. So don't let that confuse you. Make sure you understand that. Position-time graphs help us sort of understand or visualize the motion of an object much more simply. For example, let's say you were to walk 6 meters forward in 3 seconds, you stop, and then you run 6 meters back. You'd have to basically draw out all the points a, b, c, d, label all the parts and things like that. What we can do is we can actually represent it much more simply on this graph here. For example, if you were to go 6 meters forward in 3 seconds on the position-time graph, you're going from 0 all the way up to 6 in 3 seconds. So you basically just go up and over 6 and then 3. So you're going to end up right over here, and then we just connect where you started from which is at the origin with just a straight line like this.

Now finally, or the second part is that you could just stop for one second. So basically from b to c, you actually haven't moved anywhere. It's hard to see on the diagram what's actually happening. But from 3 to 4, if you're not moving anywhere, then your position just stays the same. You start at 6 and then you stay at 6. So it means we're just going to connect these points with a straight line. And then finally, you're going to go from 4 seconds all the way back to 5. So that just means from 4 to 5 over here, you're going all the way back to where you started from. So then we can label the pieces here a, b, c, and d. This graph here just represents everything you did on this diagram, but all kind of just in one place without having to draw all these little things here, and it kind of looks weird.

What the position-time graphs also do is, we can calculate the velocity directly from the graph. For example, we know that the equation is delta x over delta t. So for instance, if I look at the interval from a to b, this little line here, I can actually break this up into a triangle and I can take a look at the legs of the triangle. The vertical piece over here actually represents my change in position because that's on the vertical axis. So you can think about this as the rise of the graph. And the horizontal leg is delta t. That's the change in time, and you can think about this as the run of the graph. So that means that my delta x over delta t on this graph is really just the rise over the run, and that's a phrase we've seen before. It's the slope. So, guys, the velocity is the slope of the position graph, and there are a couple of things you want. When you have upward slopes like we did from a to b, that just means that you're moving forward and that makes sense because that represents this motion here when we move forward. If you have a horizontal slope, basically, a flat one like this, then that means that you've stopped because your position isn't changing. And then, finally, a downward slope like this from c to d is when you're moving backwards like we were moving in that last section over here.

Guys, that's all you need to know. Basically, there's just one simple equation we're looking at. It's a bunch of slopes, calculating some slopes. So let's just get to an example. So we've got this position-time graph here. I'm going to calculate the typical velocities for these parts. Let's check it out. So, we've got to go from 0 to 2 seconds. All you have to do is just go from 0 to 2 and we just need to identify those points on the graph. So at 0, I'm over here and at 2, I'm over here and I'm just going to calculate the slope of this line. So let's go ahead and do that. My average velocity is just delta x over delta t. So on the graph, I'm going to end at 10, but I'm going to start at negative 10. So that means that the delta x is my final position which is 10 minus my initial position which is negative 10. So this is my initial, that's my final, and my delta x is just equal to 20. So I've got 20 and then the time is just from 0 to 2 seconds, so that's just 2. So that means just to get a positive 10 meters per second for the velocity here.

Alright. Let's move on. So the second part is going to be from t equals 2 to t equals 4. Same thing. We're just going to go from t equals 2 to t equals 4. The line's already drawn for us. Now we just have to calculate the slope. So my average velocity here is just going to be delta x over delta t. So what's the rise? Well, if you notice here, we've got a flat line. So that means that the rise is 0. You haven't moved anywhere. You started off at 10 and you ended up at 10. So you haven't actually moved anywhere. There's no rise. So that means that 0 over whatever the time is, in this case, delta t equals 2 seconds, doesn't matter because you're still going to end up with a velocity of 0 and that makes sense because 0 meters per second means that you should have a horizontal or a flat slopeibe like this.

And then, you know, when we got 10 meters per second, we actually got an upward slope so it also makes sense. Now, for the last one, t equals 45. So this is just going to be this section right here from t equals 45. So my delta x, my rise is just going to be well, I'm going from 10 and I'm going to en...

The position-time graph for a moving box is shown below.

(a) What is the box's average velocity from 0 to 5s?
(b) What is the box's average velocity from 0 to 8s?
(c) What is the box's velocity in the interval where it's moving fastest?

• A. (a) 0 m/s; (b) 0 m/s; (c) 4 m/s
• B. (a) 4.4 m/s; (b) 3 m/s; (c) 11 m/s
• C. (a) -1.2 m/s; (b) 3 m/s; (c) 4 m/s
• D. (a) -1.2 m/s; (b) 0 m/s; (c) -11 m/s

Curved Position-Time Graphs & Acceleration

Hey, guys. So sometimes you're going to see position time graphs that are curved like this instead of just a bunch of straight lines. In this video, I'm going to show you the differences between these two types of position graphs because there are a few conceptual points you need to know. Let's check it out. So, guys, whenever you see a position time graph that is curved, not a bunch of straight lines, then what that means is that the velocity is changing. And so what that means is that the acceleration is not equal to 0. Let's check it out. When we have these kinds of straight lines in position graphs, basically, what that means is that between any two sections, the velocity is constant. So if I take the slope between this line, between any two points on this line here, I'm just going to get the same value, which means that the acceleration is equal to 0. But when you have curvature in your position graphs, when these things look like squiggly lines or curves, what that means is that the acceleration is not 0. And that's because if you take any two points on this graph, for instance, these two points, then we can see that the slope is constantly going to be changing. It's going to go from here to here to here. And so the slope or the velocity is changing means that there is some acceleration. So there are a couple of things you need to know about this acceleration. The first one is the sign of the acceleration, and there's a really simple rule. Whenever the position time graph is curving up, like a smiley face like this. Then what that means is that the acceleration is positive. One way you can think about this is that the slope over here in this first half is downwards which means that's going to be a negative velocity, just a sign. And then from here to here in the second half, now, the velocity is positive. So we have the velocity that goes from a negative number to a positive number; that can only happen when there is positive acceleration. And then basically, the opposite is true for the other type of graph. So when you have curvature that's downwards, like a frowny face like this, then what that means is that the acceleration is negative and it's basically the opposite reasoning. First, it's going positive and then the velocity becomes more negative. Now the other thing that you need to know about curves is that we can always split curves sort of down the center like this. Curves are always going to have a left side and a right side. So in this left side here, the object is always going to be slowing down. And that's because if you think about this, this is the velocity in this first section here is going to be steep. It's going to be really, really vertical. And then when you get over here, the velocity starts becoming more flat, more horizontal like this. So you're going to be slowing down because your velocity gets closer to 0. On the opposite side, on the right side of the curves, no matter which one it is, whether it's a smiley or frowny, the object is going to be speeding up. And it's because now the slope of this line, the slope between any two points is now going to get more and more vertical, which means the magnitude gets bigger. Alright, guys. That's really all you need to know about these kinds of graphs. Let's move on.

Position-Time Graphs and Instantaneous Velocity

Hey, guys. So we've already seen situations in which the position-time graph is curved like this, which means the velocity is changing. Now we already know how to calculate the average velocity and how to calculate them. And the big difference, guys, is that the average velocity is always calculated between two points whereas the instantaneous velocity is always calculated at one particular point or at one particular instant. That's why we call it instantaneous. For example, let's say we wanted to calculate the average velocity between 0 and 3 seconds over here on this graph. Then, basically, we just need a line that connects these two points from 0 to 3. So the line connecting them looks like this, and the slope of this line here is going to be our average velocity. So now what if I wanted to calculate the velocity at a particular instant? Let's say t equals 3. So I need to calculate the slope of a single point, but how do I do that? Well, it turns out I need a new kind of line called a tangent line and the tangent line looks a little bit like this. So this is the tangent line. It's a line that touches the graph only once. And one way I like to think about the tangent line is if I were to just trace along this graph over here with my pen without lifting it, and then when I get to this point, instead of following the curvature of the graph, basically keep on going in a straight line like this, that would be the tangent line. So then what happens is the instantaneous velocity is going to be the slope because the velocity is always going to be a slope, but it's going to be the slope of this tangent line that I've made. So let's take a look at the difference here. When I've got the average velocity, it's the slope between two points so I can always figure out the rise over the run. But what's tricky about the instantaneous velocity and this tangent line here is that we actually had to draw it ourselves. So a lot of problems aren't going to give you what this tangent line looks like. So you're always just going to use an approximated line. So basically, we're just using like a best guess or like a rough estimate as to what this instantaneous velocity will be. Anyway, that's really all there is to it guys. So let's take a look at an example. So we've got this position-time graph over here and let's take a look at part a. Part a is asking us to calculate the velocity between 10 and 25 seconds. So the first thing is, what kind of velocity are we talking about? Well, this is a velocity that is between two points in time. So that means it's going to be an average velocity, which means we need the slope ∆x over ∆t of the line that connects these two points. Well, at t equals 10 and 25, those are my two points right here. So this point is at t equals 10 and then this point over here is at t equals 25. So I draw the line that connects them. That's going to be this line over here and then this is my average velocity. It's the slope of this line over here. So I need you to make this triangle my rise over my run. My ∆x is going to be my final position minus my initial position. I end up at 60 over here and I started off at 30. So that means that my ∆x is 60 minus 30, which is just 30. And then over here, my ∆t is 25 minus 10 which is 15 seconds. So that means that my ∆x over ∆t is 30 over 15 and that's 2 meters per second. Let's move on to part b. Now part b is asking us to calculate the velocity at t equals 10. So now that's actually one point that they're giving us, which means this is an instantaneous velocity here. So I'm going to write v instantaneous at t equals 10. So first, I need to figure out what the slope of the line looks like, which is going to be the tangent line at t equals 10. So we're still going to look at this point over here, that's t equals 10. Now, we have to draw the tangent line. So again, we're going to trace along this graph here and when I get to this point, instead of following the curvature, I'm going to keep going in a straight line like this. So that's going to be the straight line and this kind of looks a little funky because it's going to overlap a little bit with the graph, but this is kind of like a best guess. So now I need to calculate the slope of this line over here. This here is my instantaneous velocity. So I need to calculate the ∆x over ∆t. So it's still going to be calculating the slope, but we're going to use these two points over here to calculate it. So even though we're calculating instantaneous line at one point, we're still going to use 2 points to calculate a rise over run. So, basically, I'm going to use this point and this point over here, so I have to calculate the ∆x over ∆t. So my final position is going to be 15. My initial position is going to be 45. So that means that my ∆x over here is 15 minus 45 and that's negative 30. So this is going to be my negative 30 and then the time, the time over here is between 5 and 15 seconds. So this ∆t is equal to 10. So that means that my instantaneous velocity here is negative 3 meters per second. And it's negative again because it's going downwards like this. So that's sort of you know, again, this just is a best guess or an approximation as to what this instantaneous velocity will be. So let's move on to part c. Calculate the velocity at t equals 5. So we're going to do the same thing now except t equals 5 is actually right over here. So we have to calculate the instantaneous velocity at this point. So I'm going to do this in green. So basically, I'm going to draw the line like this and when I get to this point, I'm going to keep going as if I were basically to go off in a straight line instead of following the curvature. So what I end up with because it's sort of here at the top of this little hill is I just end up with a straight line. So v when t equals 5 is going to be ∆x over ∆t. But if we'll notice that a straight line, we know from previous videos, the ∆x for a straight line is just 0 and it doesn't matter what this ∆t is. So let's say I already used these two points over here, ∆t is 10 seconds. The velocity is 0 meters per second. So what that means guys is that the velocity is equal to 0 at the peaks and the valleys of your position-time graph. Anytime you have a peak like this, the top of a hill, or anytime you have the bottom of a valley like this, there's going to be a moment in which the instantaneous velocity is flat and so therefore, it's 0. Alright, guys. That's what we need to know for this video. Let me know if you have any questions.

The position-time graph for a ball on a track is shown below.
(a) What is the ball's velocity at 4s?
(b) At what time(s) is the ball approximately travelling at -10m/s?
(c) From t = 3 to 7s, what is the sign of the acceleration?

• A. (a) +5 m/s; (b) t=7 s; (c) negative
• B. (a) +5 m/s; (b) t=8 m/s; (c) positive
• C. (a) +1 m/s; (b) t=6 m/s; (c) negative
• D. (a) +1 m/s; (b) t=3.5 m/s; (c) positive

Here’s what students ask on this topic:

What is a position-time graph and how is it used to determine velocity?

A position-time graph illustrates an object's position over time, with the y-axis representing position and the x-axis representing time. The slope of the graph indicates the object's velocity. An upward slope signifies forward motion, a horizontal line indicates a stop, and a downward slope represents backward movement. To determine velocity, you calculate the slope of the graph using the formula:

$\frac{\Delta x}{\Delta t}$

where $\Delta x$ is the change in position and $\Delta t$ is the change in time. This slope gives the average velocity over a time interval.

How do you calculate average velocity from a position-time graph?

To calculate average velocity from a position-time graph, you need to determine the slope of the line connecting two points on the graph. The formula for average velocity is:

$\frac{\Delta x}{\Delta t}$

where $\Delta x$ is the change in position (final position minus initial position) and $\Delta t$ is the change in time (final time minus initial time). This slope represents the average velocity over the selected time interval.

What does a curved position-time graph indicate about an object's motion?

A curved position-time graph indicates that the object's velocity is changing, which means there is acceleration. If the curve is upward (like a smiley face), the acceleration is positive. If the curve is downward (like a frowny face), the acceleration is negative. The changing slope of the curve shows that the velocity is not constant, and the object is either speeding up or slowing down depending on the direction of the curve.

How do you determine instantaneous velocity from a position-time graph?

To determine instantaneous velocity from a position-time graph, you need to find the slope of the tangent line at a specific point on the graph. A tangent line touches the graph at only one point and represents the velocity at that instant. The slope of this tangent line is calculated using the formula:

$\frac{\Delta x}{\Delta t}$

where $\Delta x$ is the change in position and $\Delta t$ is the change in time. This slope gives the instantaneous velocity at that specific point.

What is the difference between average velocity and instantaneous velocity?

Average velocity is calculated over a time interval between two points on a position-time graph, using the formula:

$\frac{\Delta x}{\Delta t}$

where $\Delta x$ is the change in position and $\Delta t$ is the change in time. Instantaneous velocity, on the other hand, is the velocity at a specific point in time and is determined by the slope of the tangent line at that point on the graph. While average velocity gives an overall rate of change, instantaneous velocity provides the rate of change at a particular moment.