Video: Velocity

In this video, we will learn how to find velocity as the rate of change of displacement, explaining the similarities and differences between speed and velocity.

16:38

Video Transcript

In this video, we are going to be learning about velocity. We will see what velocity actually is as well as how it can be used to understand an object’s motion.

So let’s begin by looking at the definition of velocity. To do this, let’s imagine that we have this object here, a ball. And this ball is moving in a straight line toward the right. Now, let’s say that for every second of time that passes, our ball moves one meter to the right. This means that we can say that our ball has a velocity, 𝑣, of one meter per second because it moves one meter every second. Now, this may sound a bit familiar. We may have already seen an object’s motion described in terms of the distance it moves in a given period of time. And we might be thinking, “hang on; isn’t that the speed of the ball, not its velocity?”

Well, let’s start by recalling that speed is defined as the distance moved by an object divided by the time taken for that object to move that distance. In this case, we can say that the speed of our ball is one meter, because it moves one meter, divided by one second because it moves one meter in one second, which is equal to one meter per second. So what’s going on here? What’s the difference between this quantity, which we’ve said is the speed of our ball, and velocity, which we’re trying to find out earlier? Well, there’re actually a couple of subtle differences. Let’s discuss those now.

Firstly, let’s recall that we said earlier that our ball was moving in a straight line from left to right. This is important because the velocity of an object is defined as the displacement of that object divided by the time taken for the object to be displaced by that amount. In symbols, we can write this as the velocity 𝑣 of an object is equal to 𝑠, the displacement of the object, divided by 𝑡, the time taken for the object to be displaced by that amount. And we can also recall that the displacement of an object is simply the shortest distance between an object’s initial and final positions. So, in this case, the object that we’re considering is our blue ball here. And its initial position was here. And its final position is here.

Now, the shortest distance between these two positions is simply the straight-line distance between them. And this is why it’s important that we said that that ball was moving in a straight line because for this reason, its displacement is simply one meter, the distance between the start and the finish point. Now, this is one of the differences between velocity, which is defined as displacement divided by time, and speed, which is defined as distance divided by time. We could have, for example, considered a blue ball that moved something like this to get to its endpoint, which we can say is here.

If we wanted to calculate its velocity, we would have to think about its displacement, which is the straight-line distance between its start and finish point, and divide it by the time taken for the entire journey. Whereas if we were trying to calculate the speed of the ball, we would have to take into account the entire distance moved by our ball. And we would’ve to think about the entire path of the ball’s journey. So that’s one of the subtle differences between speed and velocity.

Now, a second subtle difference comes down to the fact that displacement is a vector quantity. This means that it has both magnitude or size and direction. Whereas distance is a scalar quantity, which means it only has a magnitude or size. A consequence of this is that speed, being a distance divided by a time, is also a scalar quantity. In other words, the speed of an object is simply the distance it moves divided by the time taken for it to move that distance. The direction in which it’s moving doesn’t matter.

However, because displacement is a vector quantity, which means it includes information about the direction in which, in this case, our ball is traveling. This means that velocity must also be a vector quantity. Because displacement has information about the direction in which the ball is moving, and so velocity must also have this information. So essentially, what this means is that velocity is equivalent to the displacement of an object divided by the time taken for the object to be displaced that amount. Or, in other words, the shortest distance between its start and finish points divided by the total time of its journey. And velocity also encodes for the fact that it’s traveling, in this case, from left to right.

And this is why it was important for us to say at the beginning of the video that our blue ball is moving from left to right, and it’s moving one meter every second. Because we can now see that the velocity of our blue ball which has a magnitude of one meter per second ⁠— it’s moving one meter every second ⁠— is actually a vector quantity. And so we should include the information that it’s moving toward the right.

Now, there’re a couple of ways to encode this information. One way is to simply write this statement. We say that the velocity of our ball is one meter per second to the right. And the other is to choose a convention that says, for example, any object moving toward the right is moving in the positive direction. And therefore, any object moving toward the left is moving in the negative direction. But that’s not important to us right now. But if we choose this convention, then we can actually say that the velocity of our blue ball, 𝑣, is equal to positive one meter per second, which means it’s moving in the positive direction, that is, toward the right. And it’s moving one meter toward the right in every second.

Now, at this point, we might be wondering that if displacement is defined as the shortest distance between two points, which is the straight-line distance between those two points, then does this mean that we can only talk about velocities in straight lines? Can we only talk about an object’s velocity when it moves solely in one direction at all times? And the answer to that is no. We can talk about an object’s velocity if it changes direction as it moves. But we must realize that if an object changes direction, its velocity changes as well. This ties in with velocity being a vector quantity.

If we imagine our blue ball once again and we say that it still moves a distance of one meter for every second of time that passes, but this time our blue ball moves downward at one meter per second. Then the velocity of this blue ball is different to the velocity of the original blue ball because the first blue ball is moving to the right whereas the second is moving downward. So even though the magnitude of the velocity of this ball is one meter per second and the magnitude of the velocity of this ball is one meter per second, the two objects have different velocities because they’re moving in different directions. And so, we understand the importance of the directionality of a vector quantity.

In fact, if we now imagine our ball to be moving along a curved path ⁠— let’s say this is a section of a circle that it’s moving along ⁠— and this ball moves one meter every second. We can still say that the velocity of the ball is constantly changing because at every point along the circle, the ball is moving in a different direction. Now, we’re not going to consider any further the motion of an object that’s moving along a curved path. But instead, we will consider the motion of an object moving back and forth along the same line because this easily allows us to use our convention that we’ve decided here. Where we’ve chosen the direction for which we can say that the object has a positive velocity. And therefore, this implies that the object moving in the opposite direction must have a negative velocity.

Armed with this information, let’s think about what happens to our ball if it flip-flops back and forth, moving left to right, right to left. Firstly, let’s just consider the ball moving toward the right, but at different speeds at different times. Let’s say first of all that within a time interval of two seconds, our ball moves a distance of four meters. And then, for the next one second, the ball just stops; it just stays stationary. And then, the ball starts moving again, and in the next two seconds, the ball moves a distance of six meters. So let’s recap. In this case, the ball is moving in a straight line from left to right. And in the first two seconds, it moved a distance of four meters. Then, for one second, it stayed stationary where it was. And then, in the final two seconds, it moved a distance of six meters.

So we’ve considered the motion of the ball over a period of two plus one plus two, that is five seconds. And over these five seconds, the ball did not move at a constant velocity because, in the first two seconds, it moved four meters. But in the next second, it stayed still. And in the final two seconds, it moved six meters. So we can find the velocity of the ball for every single stage of its journey. So let’s first start by finding the velocity of our ball in its first stage of motion. Let’s call this 𝑣 subscript zero to two because this is the velocity of the ball between zero and two seconds. That’s the first two seconds of its journey.

Well, this velocity is toward the right. So we use the same convention as before that velocities toward the right are positive. And hence, we can say that the velocity of our ball between zero and two seconds is positive. And then, we can calculate that this velocity is equal to its displacement. So that’s the straight-line distance between its start point and its finish point for that stage of its journey, which happens to be four meters. And we divide this by the time taken for that ball to be displaced that amount. That’s two seconds, so we put two seconds in our denominator. Because, remember, the velocity, of an object is equal to its displacement ⁠— we’ll call this 𝑠 ⁠— divided by the time taken for the object to be displaced by that amount. And of course, because velocity is a vector, we have to account for the direction in which it moves, which we have done with our positive sign.

So anyway, the velocity of our ball then ends up being positive two meters per second in the very first stage of its motion, which means that we can move on to calculating the velocity of our ball in the next stage of its motion when it’s actually stationary. And we’ll call this velocity 𝑣 two to three because this is the velocity of the ball between two and three seconds after it started moving from this position here. Now, that velocity ends up being the displacement of our ball, which in that second was actually zero meters because remember it was stationary. It stayed exactly where it was. And because it’s not moving at all, we can’t say that it has a positive or a negative velocity because it’s not moving in a particular direction. And then, we take the displacement of the ball and divide it by the time, which was one second because the ball was stationary for a second.

Now, zero divided by another number is still zero, which means that the velocity of our ball between two and three seconds after it started its journey is zero meters per second. It wasn’t moving, kind of like what we’d expect to find, because if the ball is not moving, then it’s not going to have a velocity. And then, we can calculate the velocity of the ball between three and five seconds after it started its journey. Well, between three and five seconds, the ball moved a distance in a straight line of six meters, so its displacement was six meters to the right. Now, since anything moving toward the right has a positive velocity and positive displacement, we start with a positive sign. And then, we say that the displacement was six meters and divide this by the time taken for the ball to be displaced by that much, which was two seconds. So we put two seconds in our denominator.

Now, six divided by two gives us a numerical value of three. And the unit once again is meters per second, which means that at this point we found the velocity of the ball for every single one of its stages of motion. For the first two seconds of its motion, the ball moved at two meters per second to the right. Then, for the next second, the ball remained stationary. And then, for the last two seconds, it moved at three meters per second to the right. So at this point, we see that we found all the different velocities of the ball for all the different stages of its motion. But sometimes, we simply care about the overall motion of the ball, from the time when it started moving to the time when it finished moving. And in situations like this, a concept known as average velocity comes into play.

The average velocity of an object is simply defined as the total displacement of that object over its full range of motion divided by the total time of that journey. So, in essence, what average velocity does is to almost assume that the object is moving at a constant velocity from its start point to its finish point and calculate what that constant velocity would have to be. Based on the fact that we’re finding its total displacement and dividing it by the total time of its journey. Imagine, for example, that instead of having a blue ball moving from left to right, we’ve got a sprinter moving from left to right. And they’re taking part in a rather unusual 10-meter sprint. In that case, what we really care about is that sprinter’s average velocity.

We don’t care that it took them two seconds to move the first four meters and that they stopped moving for the next second for some weird reason. And that they ran really quickly in the last two seconds because they covered six meters. What we really care about is their average velocity, the velocity they would’ve gone out if they were covering the same amount of ground every second. They ran the same distance over the same period of time. Well, technically, in the case of the sprinter, what we really care about is the time taken to run what is actually a fixed displacement. So in this case, we care that it took them two plus one plus two ⁠— that’s five seconds ⁠— whereas somebody else may have run it quicker or slower.

Still, the point stands though. In many cases, what we care about is average velocity and not the velocity of the object along different parts of its journey. So in this case, we can calculate the average velocity of our blue ball by saying that 𝑣 subscript avg — that’s the average velocity. Is equal to the total displacement of the blue ball, which ends up being four meters plus zero meters when it was stationary plus six meters, divided by the total time which ends up being two seconds plus one second plus two seconds. And of course, because all of these velocities are toward the right, we must include a positive sign to let us know that the average velocity is going to be positive.

So evaluating this fraction, we find it to be positive 10 meters in the numerator divided by five seconds in the denominator. We find the numerical value to be 10 divided by five, which is two. And the unit is meters divided by seconds or meters per second. And hence, we find that if the blue ball had moved at a constant velocity covering the same displacement in the same total time, it would’ve moved at a velocity of two meters per second to the right. And as we’ve already mentioned, average velocity gives us more of a sense of the overall motion of our object, rather than breaking it up into individual stages of motion. And actually, average velocity becomes even more useful when we consider velocity in a negative direction.

Let’s say now that our ball starts out here. And let’s say in the first two seconds, it once again moves a distance of four meters to the right. And then, once again when it reaches this point, for a period of one second, the ball remains stationary. It stays exactly where it is. But then, in the next two seconds, the ball reverses its direction of motion. And let’s say it travels six meters toward the left this time. And so what we’re saying is that it takes another two seconds to travel a distance of six meters to the left, which means that our ball started out here and ended up here.

Now, we can conduct the same analysis as before. Where we find the velocity of the ball along each one of its stages. And we can see that the velocity of the first stage, velocity from zero to two seconds, will be the same as it was earlier because it moved in the positive direction toward the right. A distance of four meters in two seconds, so that gives a velocity of positive two meters per second. And the velocity during the stationary phase, so that’s from two to three seconds, will be the same as before, zero meters per second. But this time, the velocity between three and five seconds will be different because, now, the ball was moving leftward. So its velocity is going to be its displacement, which is negative six meters, divided by the time taken for that displacement to occur, which was two seconds. And so its velocity in this phase would be negative six divided by two meters per second or negative three meters per second.

But if what we really care about is the overall motion of our ball, then the important thing is finding its average velocity, which once again is equal to the total displacement of the ball divided by the total time of its journey. So what is the total displacement of the ball? Well, it’s the shortest distance between its start point at the beginning of its journey and its endpoint, that is, at the end of the journey. And we can recall that it started here and finished here, regardless of how it got there. In this case, it went toward the right first then stayed stationary at this point and then moved back toward the left. The ball’s displacement is still this length here and it’s toward the left because this is the start; this is the finish. So it had to move toward the left.

Now, this distance here is simply six meters minus four meters, technically, four meters plus negative six meters. And so we can say that the average velocity of our ball in this scenario is equal to its displacement, which happens to be negative two meters — it moved two meters to the left from its start to its finish — divided by the time taken for its entire journey, which happened to be five seconds once again. Because for the first two seconds, it moved right. Then, it stayed stationary for one second. And then, it moved left in the final two seconds. All together, that’s five seconds. And so we find that the average velocity of our ball in this situation is negative two-fifths meters per second. Or, in decimal form, it’s negative 0.4 meters per second, which is essentially telling us that the entirety of our ball’s motion is equivalent to if it had just moved toward the left at a constant 0.4 meters per second.

So having learnt about velocity as well as average velocity, let’s summarize what we’ve talked about in this lesson. First of all, we saw that velocity is defined as the displacement of an object divided by the time taken for the object to be displaced by that amount. We saw that, in symbols, this can be written 𝑣, the velocity, is equal to 𝑠, the displacement, divided by the time 𝑡. We also saw that velocity is a vector quantity. It has both magnitude and direction. And finally, we saw tht average velocity was defined as the total displacement of an object divided by the total time of its journey. We saw that it’s useful for analyzing the overall motion of objects that are not moving at a constant velocity. And this is a summary of velocity.

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