Lesson Explainer: Relative Velocity | Nagwa Lesson Explainer: Relative Velocity | Nagwa

# Lesson Explainer: Relative Velocity Mathematics

In this explainer, we will learn how to calculate the relative velocity of a particle with respect to another and how to calculate a relative velocity vector.

When a particle moves in a straight line from one point, , to another, , we can describe its displacement using the vector

If the particle moves from to in the time interval , then the velocity of this particle is given by

This velocity is called the average velocity of the particle as it moves from to . If the velocity of the particle is constant, then this average velocity is equal to the velocity of the particle at any point during its motion. There are some situations where we may want to consider a different kind of velocity, which we call the relative velocity.

Suppose you are sat on a train, looking out the window at a car driving in the opposite direction. Suppose also that the car is moving with a constant speed of 50 km/h and the train is moving with a constant speed of 80 km/h. Using this, we can say that the velocity of the train is 80 km/h, while the velocity of the car is km/h. However, since you are sat on the train, which is moving at 80 km/h, while observing the car move in the opposite direction, it will appear as though the car is moving faster than its speed of 50 km/h. This is due to relative velocities.

Relative velocities tell us that movement is a relative concept, which means that it differs depending on the observer. In every case, while the viewer observes the movement of the other object, they consider themself to be at rest even if they are not. The velocity that the viewer observes the object moving at is not the actual velocity of the object, it is the relative velocity.

Let us now consider how we can calculate relative velocities. Suppose that we have two bodies, and , such that the velocity of is and the velocity of is . A stationary observer would see the two bodies moving with their velocities and .

Now, let us suppose the observer is sat on body , moving with velocity . The observer assumes that they are stationary, so when we are finding the relative velocity of , we will need to take this into consideration.

First, suppose that , so is stationary. The observer is moving away from with velocity , so they will see move away from them with velocity . This is the relative velocity of with respect to , which we can call . In the case where ,

Next, we will consider when . Since the observer is still moving away with velocity , we will again need to take this into consideration. This time, we also have . To find the relative velocity of with respect to , we need to simply add to the we had before. This gives us that the relative velocity is

### Definition: Relative Velocity

If bodies and are moving with velocities and , respectively, then we can say that the relative velocity of with respect to , , is given by and the relative velocity of with respect to , , is given by

These formulas are given in terms of vectors, but they still hold for scalar quantities too.

Let us look at an example that considers relative velocity purely in terms of the differences between velocity vectors.

### Example 1: Finding Relative Velocity Using Unit Vectors

Fill in the blank: If and , then .

The difference between and is given by

Rearranging to find gives

Substituting known values gives

Now, let us look at another such example.

### Example 2: Finding Relative Velocity Using Unit Vectors

If and , then the relative velocity .

is a unit vector in some fixed direction.

The difference between and is given by

Substituting known values gives

Let us consider an example where there is a context supplied, involving a body that measures the velocity of a second body relative to the first body.

### Example 3: Relative Speed of Bodies Moving in Opposite Directions

A car is moving on a straight road at 84 km/h, and in the opposite direction, a motorbike is moving at 45 km/h. Suppose that the direction of the car is positive. Find the velocity of the motorbike relative to the car.

Let be the velocity of the car and be the velocity of the motorbike. The direction of the car is positive; hence, and

The velocity of the motorbike relative to the car is given by

If two bodies move one-dimensionally in opposite directions, their speeds are added to determine the speed of either body relative to the other. Let us look at another example where this occurs.

### Example 4: Finding the Time to Complete a Journey Using Relative Velocities

A ship was sailing with a uniform velocity directly toward a port that is 144 km away. A patrol aircraft passed over the ship traveling in the opposite direction at 366 km/h. When the aircraft measured the shipβs speed, it appeared to be traveling at 402 km/h. Determine the time required for the ship to reach the port.

To determine the time required for the ship to reach the port, it is necessary to know the speed at which the ship approaches the port. The port is assumed to be stationary.

The speed of the ship measured by the aircraft is 402 km/h. As the ship and aircraft travel in opposite directions, 402 km/h is the sum of their speeds. The speed of the aircraft is stated to be 366 km/h, so the speed of the ship is given by

The time required to travel 144 km at a speed of 36 km/h is given by

Let us consider an application of relative velocity in a context involving two bodies moving in the same direction. For two bodies moving in the same direction at speeds and , respectively, the speed of either body relative to the other, , is given by

The trivially obvious case of this is the case corresponding to both bodies having the same speed, and hence the position of one body relative to the other is constant throughout the motion of the bodies.

### Example 5: Using Relative Velocity to Find the Length of a Train given the Time Taken by a Moving Object to Pass It

A helicopter flew in a straight line at 234 km/h above a train moving in the same direction. It took the helicopter 21 seconds to travel the length of the train. Following this, the pilot halved the helicopterβs speed. Given that it took the train 14 seconds to pass the helicopter traveling at this speed, find the length of the train in metres.

The most important thing to appreciate in this question is that because the helicopter and the train move in the same direction throughout, relative to the ground, their velocities have the same sign. The difference in their velocities is thus equal to the difference in their speeds, and the speed of the train relative to the helicopter (and vice versa), , is simply the difference between their speeds.

In the first 21 seconds, the helicopter has a greater velocity relative to the ground than the train has, and in the 14 following seconds, the train has a greater velocity relative to the ground than the helicopter has. The change in the velocity of the helicopter between the first and the second time intervals is assumed to occur in negligible time.

In each time interval, it is the case that where is the distance that the helicopter moves relative to the train (and vice versa), which is also the length of the train. The difference between the time intervals allows us to determine the velocity of the train.

To simplify finding the length of the train in metres, the speed of the helicopter is converted to a speed in metres per second as follows:

For the first time interval,

For the second time interval,

The length of the train remains constant, so the two terms can be equated to give

This can be rearranged to determine the velocity of the train. Both sides of the equation can be divided by 14 to give

The bracket can be expanded to give

The rearrangement is then completed as follows:

This value for can now be substituted into the equation for in either time interval. If the first of the time intervals is used, this gives us

If the second of the time intervals is used, this gives us

The length of the train is 273 metres.

### Key Points

• If bodies and are moving with velocities and , respectively, then we can say that the relative velocity of with respect to , , is given by and the relative velocity of with respect to , , is given by
• When finding relative velocities, it is useful to establish a sign convention for solving problems.