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# Relative velocity in two dimensions

## Relative Velocity:

We encounter occasions where one or more objects move in a frame which is non-stationary with respect to another observer. For example, a boat crossing a river that is flowing at some rate or an aeroplane encountering wind during its motion. In all such instances, to describe the object’s complete motion, we need to consider the effect that the medium is causing on the object. While doing so, we calculate the relative velocity of the object considering the velocity of the particle as well as the velocity of the medium. Here, we will learn how to calculate the relative velocity.

## What is Relative Velocity?

Let us consider two objects, A and B moving with velocities Va and Vb with respect to a common stationary frame of reference, say the ground, a bridge or a fixed platform.

The velocity of the object A relative to the object B can be given as,

$$\begin{array}{l}V_{ab}=V_{a}-V_{b}\end{array}$$

Similarly, the velocity of the object B relative to that of object a is given by,

$$\begin{array}{l}V_{ba}=V_{b}-V_{a}\end{array}$$

From the above two expressions, we can see that

$$\begin{array}{l}V_{ab}=-V_{ba}\end{array}$$

Although the magnitude of both the relative velocities is equal to each other. Mathematically,

$$\begin{array}{l}\left |V_{ab} \right |=\left |V_{ba} \right |\end{array}$$

## Examples of Relative Velocity

We can understand the concept of relative velocity more clearly with the help of the following example.

Example: A plane is travelling at velocity 100 km/hr, in the southward direction. It encounters wind travelling in the west direction at a rate of 25 km/hr. Calculate the resultant velocity of the plane.

Given, the velocity of the wind = Vw = 25 km/hr

The velocity of the plane = Va= 100 km/hr

The relative velocity of the plane with respect to the ground can be given as

The angle between the velocity of the wind and that of the plane is 90°. Using the Pythagorean theorem, the resultant velocity can be calculated as,

R2= (100 km/hr)2 + (25 km/hr)2

R2= 10 000 km2/hr2 + 625 km2/hr2

R2= 10 625 km2/hr2

Hence, R = 103.077 km/hr

Using trigonometry, the angle made by the resultant velocity with respect to the horizontal plane can be given as,

$$\begin{array}{l}tan\theta =(\frac{wind \: velocity}{aiprplane\: velocity})\\ \\ tan\theta =(\frac{25}{100})\\ \\ \theta =tan^{-1}\frac{1}{4}\\ \\ \theta =14.0^{\circ}\end{array}$$

## Relative Velocity Problems

1) What is relative velocity?

Relative velocity is defined as the velocity of an object B in the rest frame of another object A.

2) A motorcycle travelling on the highway at a velocity of 120 km/h passes a car travelling at a velocity of 90 km/h. From the point of view of a passenger on the car, what is the velocity of the motorcycle?

Solution:

Let us represent the velocity of the motorcycle as VA and the velocity of the car as VB.

Now, the velocity of the motorcycle relative to the point of view of a passenger is given as

VAB = VAVB

Substituting the values in the above equation, we get

VAB = 120 km/h – 90 km/h = 30 km/h

Hence, the velocity of the motorcycle relative to the passenger of the car is 30 km/h.

3) A swimmer swimming across a river flowing at a velocity of 4 m/s swims at the velocity of 2 m/s. Calculate the actual velocity of the swimmer and the angle.

Solution:

The actual velocity of the swimmer can be found out as follows:

$$\begin{array}{l}V_{actual}=\sqrt{2^2+4^2}=4.47\,m/s\end{array}$$

The angle is calculated as follows:

$$\begin{array}{l}\tan \theta =\frac{2}{4}\end{array}$$

$$\begin{array}{l}\theta =\tan^{-1}\frac{2}{4}=26.57^{\circ}\end{array}$$

4) A person in an enclosed train car, moving at a constant velocity, throws a ball straight up into the air in her reference frame.

• At what point, does the ball land?
• Where does the ball land if the car slows down?
• Where does the ball land if the car speeds up?
• Where does it land if the car rounds a turn?

Solution:

1. The ball lands at the point from which it was thrown, i.e. back to the thrower’s hand.
2. It lands in front of the point from which it was thrown.
3. It lands behind the point from which it was thrown.
4. The ball will land to the left of the point from which it was thrown if the car takes a right turn and vice versa.

5) An aeroplane A flies with a velocity of 450 m/s to the north, while an aeroplane B travels at a velocity of 500 m/s to the south beside aeroplane A. Calculate the relative velocity of the aeroplane A with respect to aeroplane B.
Solution:

The relative velocity of aeroplane A with respect to the velocity of aeroplane B is calculated as follows:

VAB = VAVB

Substituting the values in the equation, we get

VAB = 450 m/s – (–500 m/s) = 950 m/s

The velocity of aeroplane B is considered negative, as it flies in the opposite direction to the of aeroplane A.

Stay tuned with BYJU’S to learn more about the concept of relative velocity, relative motion and other related topics.

## Frequently Asked Questions – FAQs

### What is relative velocity?

The relative velocity is defined as the velocity of an object with respect to another observer.
It is the time rate of change of relative position of one object with respect to another object.

### A motorcycle travelling on the highway at a velocity of 120 km/h passes a car travelling at a velocity of 90 km/h. From the point of view of a passenger in the car, what is the velocity of the motorcycle?

Let us represent the velocity of the motorcycle as VA and the velocity of the car as VB.

Now, the velocity of the motorcycle relative to the point of view of a passenger is given as, VAB = VA – VB

Substituting the values in the above equation, we get VAB = 120 km/h – 90 km/h = 30 km/h

Hence, the velocity of the motorcycle relative to the passenger of the car is 30 km/h.

### State if the statement is true or false: Relative velocity can be negative.

The above statement is true. Relative velocity can be negative. As relative velocity is the difference between two velocities irrespective of their magnitude, it can be negative.

### What is the difference between velocity and relative velocity?

The difference between velocity and relative velocity is that velocity is measured with respect to a reference point which is relative to a different point. While relative velocity is measured in a frame where an object is either at rest or moving with respect to the absolute frame.

### What is the need of using relative velocity?

The need for using relative velocity is that it is used for differentiating if the object is at rest or moving.

Test your knowledge on Relative Velocity

#### 1 Comment

1. Sourav

Two bodies A and B are moving in the same direction with velocities VA
and VB
, where VA>VB
and
they start symultaneously from different points, and B is in front of A. Represent their relative motion
in a position time graph. Explain their relative velocity during the motion