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

There are two ways that we can understand the speed of an object. These are

- its absolute speed,
- its relative speed.

Let us first discuss absolute speed. This is often just called speed. In this explainer, where “speed” is written without stating if the speed is absolute or relative, it means “absolute speed.”

The speed of an object equals the distance that an object moves in a certain time.

An object moves from one particular position to another. An example of this would be a runner moving from the start line to the finish line of a racetrack. This is shown in the following figure.

For the example of a runner moving from the start line to the finish line of a racetrack, we see that

- the initial position of the runner is the position of an object, the start line,
- the final position of the runner is the position of an object, the finish line.

These initial and final positions are labeled in the following figure.

A speed does not depend on what objects’ positions are used, which could be the positions of any objects we like. We could even just name the initial position “position A” and the final position “position B.”

The factors that determine a speed are

- the distance an object moves from one position to another,
- how much time passes while the object moves.

Speed can be represented by the formula

This formula assumes that the initial and final positions of the object do not change when the object moves. This means that changes in the distance between an object and either its initial or final position happen only because of the motion of the object.

We have now clarified what we mean by absolute speed.

Let us now discuss the relative speed of an object.

For relative speed, the initial and final positions of an object can change while the object moves.

We can consider two examples to show how a relative speed can be either the same as or different to an absolute speed.

In the first example, a relative speed is the same as an absolute speed.

Let us now consider the first example.

In this example, the following figure shows two cars that are initially next to each other.

We see that the red car does not move. Its position is the same at 0 seconds and at 1 second.

The green car does move, however, and so there is an increase in the distance between the cars. The increase in the distance between the green car and the red car in a time of 1 second is shown by the black arrow.

We will consider the speed of the green car relative to the red car.

This means that we determine how much the distance between the red car and the green car changes in a time of 1 second.

Using the formula we see that the speed of the green car relative to the red car is given by

The increase in the distance between the cars is equal to the distance moved by the green car. This is because the increase in distance between the cars is only due to the motion of the green car.

We see that in this case, the speed of the green car relative to the red car is the same as the absolute speed of the green car.

Now, let us consider the second example where a relative speed is different to an absolute speed.

In this example, the red car does move. This is shown in the following figure.

The cars are again initially next to each other. The change in distance between the cars in this example though is due to the motion of both cars, not just the green car.

The speed of the green car relative to the red car is still given by but the change in the distance between the cars is much less now than in the first example. The black arrow showing this distance is shorter than that in the first example.

The change in distance is different in this example because the movement of both the green car and the red car affects the change in distance, not just the motion of the green car.

We see then that the speed of the green car relative to the red car is different to the absolute speed of the green car if the red car moves.

Let us now look at an example in which the change in the distance between moving objects is considered.

### Example 1: Determining the Rate of the Change in Distance between Two Moving Objects

A blue object and an orange object move across a grid of lines spaced 1 metre apart. Each object moves for 2 seconds. The arrows show the distances that the objects move in each second. By how much does the distance between the objects change each second?

- 0 m
- 1 m
- 2 m

### Answer

Let us see what we can determine about these objects from the diagram of their motion.

The arrows that represent the motion of the objects point in the same direction.

We can see that both objects move in the same direction, to the right.

The arrows represent the distance moved by an object to the right in a time of 1 second.

We can see that the arrows for each object are the same length. We can see that the length of the arrows equals the distance between adjacent grid lines. The question states that the grid lines are spaced 1 metre apart.

We see then that each object moves 1 metre in 1 second. The absolute speed of each object is then 1 metre per second.

The question asked us though, by how much the distance between the objects changed each second.

This tells us that we are not trying to find the absolute speeds of the objects. It is though useful to have determined that the objects have the same absolute speed as each other.

The following figure shows the positions of the two objects at 1-second time intervals.

We see that each second, the distance between the objects increases by 1 metre due to the motion of the blue object and the distance between the objects decreases by 1 metre due to the motion of the orange object.

The change in distance, , each second is then given by

To answer the question of how much the distance between the objects changes each second, the answer is metres.

Let us again consider the example of the two moving cars. This is shown in the following figure.

Recall that, in this example, we said that we determined the speed of the green car relative to the red car.

Another way of saying this is that we determined the speed with which the green car moved away from the red car.

We could just as well have said that we determined the speed with which the red car moved away from the green car.

This seems strange, as we know that the red car moved in the same direction as the green car.

It seems to make sense to say that the green car traveling a greater distance from its initial position than the red car means that the green car was moving away from the red car.

It does not seem to make sense to say that the red car traveling a lesser distance from its initial position than the green car means that the red car was moving away from the green car.

We should make sure though that we understand that “moving away from” means only “increasing the distance between.”

We know that the distance between the cars increases.

We also know that the distance between the cars is the same whether we call this distance “the distance from the green car to the red car” or “the distance from the red car to the green car.”

The length of the line showing the distance between the cars is the same whether we measure it left to right or right to left.

This means that the speed of the green car relative to the red car is equal to the speed of the red car relative to the green car.

We can say the speed of each car relative to the other car is the same.

Let us now look at an example in which the relative speed of moving objects is considered.

### Example 2: Determining the Relative Speed of Two Moving Objects

A blue object and an orange object move across a grid of lines spaced 1 metre apart. Each object moves for 2 seconds. The arrows show the distances that the objects move in each second. What is the speed of either object relative to the other?

- 0 m/s
- 1 m/s
- 2 m/s

### Answer

Let us see what we can determine about these objects from the diagram of their motion.

The arrows that represent the motion of the objects point in the same direction.

We can see that both objects move in the same direction, to the right.

The arrows represent the distance moved by an object to the right in a time of 1 second.

We can see that the arrows for each object are the same length. We can see that the length of the arrows equals the distance between adjacent grid lines. The question states that the grid lines are spaced 1 metre apart.

We see then that each object moves 1 metre in 1 second. The absolute speed of each object is then 1 metre per second.

The question asked us though not what the absolute speeds of the objects were, but what the speed of either object relative to the other was.

The following figure shows the positions of the two objects at 1-second time intervals.

We see that each second, the distance between the objects increases by 1 metre due to the motion of the blue object and the distance between the objects decreases by 1 metre due to the motion of the orange object.

The change in distance, , each second is then given by

Speed is calculated using the formula

The speed of the blue object relative to the orange object is the number of metres of distance the blue object moves away from the orange object in 1 second.

The distance between the objects changes by metres each second. The speed of the blue object relative to the orange object is then zero.

We can say the same thing about the speed of the orange object relative to the blue object.

The speed of the orange object relative to the blue object is the number of metres of distance the orange object moves away from the blue object in 1 second. The distance between the objects changes by metres each second. The speed of the orange object relative to the blue object is then zero.

To answer the question of what the speed of either object relative to the other is, the answer is metres per second.

We see from this that two objects that move in the same direction at the same speed have a speed of zero relative to each other. This would be true whatever absolute speed the objects had, providing the absolute speed was the same for both objects.

We have seen how to determine the relative speed of objects that move in the same directions.

We can also consider how to determine the relative speed of objects that move in opposite directions.

One example of this, shown in the following figure, is if the objects move toward each other.

Another example of this, shown in the following figure, is if the objects move away from each other.

In both cases, what is important is how much the distance between the objects changes per second.

The speed of either object to the other is the number of metres that the distance between them changes per second.

Let us now look at an example in which the relative speed of moving objects is considered.

### Example 3: Determining the Relative Speed of Two Objects Moving Toward Each Other

A blue object and an orange object move across a grid of lines spaced 1 metre apart. Each object moves for 2 seconds. The arrows show the distances that the objects move in each second. What is the relative speed at which the objects approach each other?

- 2 m/s
- 1 m/s
- 4 m/s

### Answer

Let us see what we can determine about these objects from the diagram of their motion.

The arrows that represent the motion of the objects point in opposite directions.

We can see that both objects move in opposite directions.

The arrows represent the distance moved by an object to the right in a time of 1 second.

We can see that the arrows for each object are the same length. We can see that the length of the arrows equals the distance between adjacent grid lines. The question states that the grid lines are spaced 1 metre apart.

We see then that each object moves 1 metre in 1 second. The absolute speed of each object is then 1 metre per second.

The question asked us though not what the absolute speeds of the objects were, but at what relative speed the objects approached each other.

The following figure shows the positions of the two objects at 1-second time intervals.

We see that each second, the distance between the objects decreases by 1 metre due to the motion of the blue object and the distance between the objects decreases by 1 metre due to the motion of the orange object.

The change in distance, , each second is then given by

Speed is calculated using the formula

The speed of either object relative to the other object is the change in the number of metres of distance between the objects in 1 second.

The relative speed at which the objects approach each other is then given by

Let us look at another example.

### Example 4: Determining the Relative Speed of Two Objects Moving Away from Each Other

A blue object and an orange object move across a grid of lines spaced 1 metre apart. Each object moves for 2 seconds. The arrows show the distances that the objects move in each second. What is the relative speed at which the objects move away from each other?

- 2 m/s
- 1 m/s
- 4 m/s

### Answer

Let us see what we can determine about these objects from the diagram of their motion.

The arrows that represent the motion of the objects point in opposite directions.

We can see that both objects move in opposite directions.

The arrows represent the distance moved by an object to the right in a time of 1 second.

The question asked us though not what the absolute speeds of the objects were, but at what relative speed the objects approached each other.

The following figure shows the positions of the two objects at 1-second time intervals.

We see that each second, the distance between the objects increases by 1 metre due to the motion of the blue object and the distance between the objects increases by 1 metre due to the motion of the orange object.

The change in distance, , each second is then given by

Speed is calculated using the formula

The speed of either object relative to the other object is the change in the number of metres of distance between the objects in 1 second.

The relative speed at which the objects move away from each other is then given by

Let us now look at an example involving familiar objects.

### Example 5: Comparing the Absolute Speed of Two Objects Using Relative Motion

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?

- The speed of the yacht is greater than the speed of the person walking.
- The speed of the person walking is greater than the speed of the yacht.

### Answer

This question can be answered by considering the initial and final positions of the person on the yacht relative to the person on the island.

Initially, the person on the yacht can be seen to be on the right of the person on the island. The person on the yacht is also moving toward the right-hand end of the yacht.

It is important to note that the question tells us that the person on the island sees the person on the yacht move in the direction of the motion of the yacht.

The diagram in the question shows that the yacht moves to the left. This is the direction in which the person on the island sees the person on the yacht move.

After sufficient time passes, the person on the yacht must be seen by the person on the island to be on their left.

The following figure shows the island and the person on the yacht at an initial time and a final time, when the person on the yacht is seen to the left of the person on the island.

At the initial time, the person on the yacht is to the right of the person on the island.

At the final time, the person on the yacht is to the left of the person on the island.

We can see that the person on the yacht has moved toward the left more than to the right.

This tells us that the speed with which the person on the yacht moves to the left must be greater than the speed with which they move to the right.

The motion of the person on the yacht to the left is due to the motion of the yacht.

We see then that the yacht must have greater speed than the person walking.

Let us now summarize what we learned in this explainer.

### Key Points

- Absolute speed is the distance moved by an object per second.
- The speed of an object relative to another object is the change in the distance between the objects per second.
- Two objects that move relative to each other have the same speed relative to each other.
- Two objects that move in the same direction at the same speed are at rest relative to each other.