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Lesson Video: Relative Speed Science

In this video, we will learn how to determine the speeds of objects relative to other objects.

12:19

Video Transcript

In this video, we will learn how to determine the speeds of objects relative to other objects.

To understand relative speed, let’s begin with speed of a different type. Say that here we have a running track with a runner moving on it. If the runner moves ahead at a speed of three meters per second, that tells us that for every second of time that passes, the runner moves down the track three additional meters. This means that the runner speed of three meters per second is three meters per second relative to this track. The track, of course, isn’t moving. And this means we can call the runner’s speed of three meters per second the runner’s absolute speed.

The absolute speed of any object is the speed of that object as it moves relative to something that doesn’t move. As we see here, the thing that doesn’t move is the track. When we talk simply about the speed of an object, often we mean its absolute speed. Compared to that moving object’s motion, say, the motion of this runner, there’s something which is not moving relative to that object. And this lets us know the moving object’s absolute speed.

Now let’s say that our scenario changes. We have the second runner moving towards the first at an absolute speed of four meters per second. That is for every one second of time this runner passes over four meters of the track. We know the speed of the first runner compared to the track is three meters per second and the speed of the second runner compared to the track is four meters per second. But what is the speed of this first runner compared to the second runner? Finding this out means solving for the relative speed of these runners.

The speed of one object relative to another can be defined mathematically like this. It equals the change in distance between those two objects divided by a change in time. Let’s apply this relationship to our runners. We’re talking about a change in distance over time. Let’s say that that amount of time is one second. We’ve seen that in one second of time this runner will move three meters along the track. Then over that same second of time — from what we’ve called zero seconds up to one second — the second runner moves a distance of four meters along the track. We know this because the second runner’s speed relative to the track is four meters every second. For our runners then, over a time interval of one second, their total change in distance is three meters plus four meters.

Now the reason we use one second as our change in time is because this second of time during which the first runner covered three meters is the same second of time for which the second runner covered four. It’s not two separate seconds, so we don’t use, say, two seconds. Rather, it all happens in one second of time and so that is our change in time. Three plus four is seven. So, the relative speed of our runners is seven meters per second.

Now here’s something interesting. This relative speed is the speed of the first runner relative to the second runner, and it’s the speed of the second runner relative to the first. Comparing either moving object to the other, the relative speed between them is seven meters per second. Later on, we’ll get some more practice using this equation.

For now, notice that we’ve called this an equation for relative speed, but that we can still use this equation, even if the two objects we’re considering aren’t both in motion. For example, if we wanted to solve for the relative speed of the second runner compared to the track, then in that case, over a time interval of one second, the change in distance between those objects would be four meters. The speed of the second runner compared to the track and the speed of the track compared to the second runner is four meters per second.

As we said, because one of these objects isn’t in motion — that is, the track isn’t moving — we call this the absolute speed of the runner. An object’s relative speed can also be its absolute speed if the thing its speed is measured relative to is not in motion. All that to say we can use this equation to solve for the speed between any two objects even if the objects aren’t both moving. Let’s now get some practice working with relative speed through some examples.

A blue object moves across a grid of lines spaced one meter apart. The object moves for two seconds. The arrows show the distance that the object moves in each second. What is the speed of the object relative to the grid lines that it crosses?

Looking at these grid spaces, we’re told that the side of each square has a length of one meter. We’re also told that the arrows — there’s one here and one here — show the distance that the object moves over each second of time. In the first second, the object moves one grid space. That’s one meter of distance. In the second second, it also moves one meter. We want to find the speed of this object relative to the grid lines. To do this, let’s recall that the relative speed between two objects equals the change in distance between them divided by the change in time.

In this scenario, our blue object is definitely in motion, while our grid lines are not; they don’t move at all. So, the only change in distance between our objects will be caused by the motion of our blue object. The amount of time over which the blue object is in motion is one second plus one second. And we saw that in each of those seconds the blue object moves one grid space or one meter. Therefore, the blue object has moved two meters in two seconds. Its speed then is one meter per second. This is the speed of the blue object relative to the grid lines. And notice since the grid lines aren’t moving at all, it’s also the absolute speed of the blue object.

Whenever we have two objects where one object is moving and one is not, the relative speed between them is the same as the moving object’s absolute speed.

Let’s look now at another example.

A blue object and an orange object move across a grid of lines spaced one meter apart. The arrows show the distances that the objects move in each second. What is the speed of either object relative to the other?

We’re told here that these grid spaces each have a side length of one meter and that the arrows show the distances moved by the orange and blue objects in each second of time. Knowing this, we want to solve for the speed of either of these objects relative to the other. To get started, let’s recall the mathematical expression for relative speed. The speed of one object relative to another is the change in distance between them divided by the change in time.

For our blue and orange objects, we’ll say that they had their original positions at a time of zero seconds. Then each object moved for one second to reach its final position. The total change in time, then, is just one second. That’s the time period over which both objects were moving. Over that time, we see the orange object moved one, two, three grid spaces; that’s three meters. Meanwhile, the blue object moved one, two grid spaces or two meters. The change in distance between these two moving objects is three meters plus two meters. We get a result of five meters per second.

Note that because these objects are approaching one another, their relative speed is how many meters closer one object gets to the other object per second. If instead they have been moving away from one another, their relative speed is how many meters away one object gets from the other in each second of time. So, five meters per second is the speed of the orange object relative to the blue object. And it’s also the speed of the blue object relative to the orange object. That is, it’s the speed of either object relative to the other.

Let’s look at another example.

A blue object and an orange object move across a grid of lines spaced one meter apart. The arrows show the distances that the objects move in each second. What is the speed of either object relative to the other?

To find the relative speed between these objects, let’s start by recalling the mathematical equation for relative speed. For two objects, the relative speed between them equals the change in distance between them divided by the change in time. So, for our orange and blue objects, we’ll first figure out how the distance between them changes. And then, we’ll see over what interval of time that change occurs.

We see that the initial position of the orange object is here and the initial position of the blue object is here. That is, at first, these objects are separated by one, two, three, four grid spaces. Each grid space is one meter of distance, so four grid spaces is four meters. Both of these objects then move and so that they reach final positions at these locations. At the end of this time interval, the orange and blue objects are one, two, three grid spaces apart. The change in distance between them then is the larger distance minus the smaller one. All this change occurred over some time interval.

We’re told that in our diagram the arrows show the distances that the objects move in each second of time. Since there’s one arrow for each object, we know that each object moved for one second. The total change in time then is just the one second over which both the orange and blue objects were in motion. Four meters minus three meters is one meter. So, the relative speed between these objects is one meter per second.

Note that to calculate this relative speed, we needed to subtract one distance from another. This happens whenever our two objects are moving in the same direction. If they had been moving apart, we could have approached the problem differently. All that said, the speed of the orange object relative to the blue and the blue relative to the orange is one meter per second.

Let’s look now at one last example.

A yacht passes a small island upon which a person is standing. A person on the yacht walks through a doorway and then along the deck in the opposite direction to the motion of the yacht. The person on the island sees the person on the yacht move in the direction of the motion of the yacht. Which is greater, the speed of the yacht or the speed of the person walking?

Looking at this picture, we see a yacht moving along to the left, while a person on the yacht walks to the right. All this is observed by a person standing still on this island. We want to know which speed is greater, the speed of the person walking on the yacht’s deck or the speed of the yacht itself. To answer this question, let’s imagine watching the yacht through the eyes of the person standing still. At one instant in time, the yacht is here and the person on the yacht is standing in the doorway. Then later, the yacht has moved over here and the person has moved to the back of the yacht. The question is, which object moves faster, the yacht in moving from here over to here or the person in moving from the doorway to the back of the yacht?

We’ve seen that while the yacht is moving to the left, the person on the yacht is walking to the right. Here’s what that means. Between these two instants in time, if the person on the yacht is walking faster than the yacht is moving through the water, then overall in the eyes of the person standing on the island, the person on the yacht will move to the right. Again, that’s what would happen if the person on the yacht was walking faster than the yacht traveled itself. On the other hand, if the yacht moves faster than the person walks, then we would expect the person overall to move to the left as the yacht moves.

The question is, which of these two outcomes do we see? At the first instant in time, the person on the yacht is here, while at a later instant that person is here. That means the person on the yacht’s overall motion is to the left, even though they were walking to the right on the yacht. We could say then that the speed of the yacht has overcome the speed of the person; the speed of the yacht must be greater. Our answer is that the speed of the yacht is greater than the speed of the person walking.

Let’s now finish up our lesson by reviewing a few key points. In this video, we learn that the absolute speed of an object is the speed of that object relative to another object that does not move. On the other hand, relative speed is the speed between two objects that may both be moving. The equation for finding relative speed is the change in distance between two objects divided by the change in time. We can use this equation whether both objects are moving, one is moving and the other is not, or even if both objects are still. This is a summary of relative speed.

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