Relative Velocity in Kinematics - Concepts with Numerical for JEE

While studying the motion of different objects, we might run into a scenario where two or more objects are in motion in a frame and the frame itself is moving with respect to some other observer. When such a condition arises, we need to consider the effects of the motion of the frame on the objects or say we need to observe the objects relative to the frame. Here originates the concept named relative velocity.

While travelling in a vehicle we often see the outside buildings and trees apparently moving backwards and the person sitting next to us in the vehicle appears stationary. But when you are standing on the ground, the trees and buildings seem stationary. It’s because you see the motions with respect to your position and your state of rest or motion. For you, your body is stationary and the other objects are moving. The velocities you are observing for the objects are nothing but the relative velocity of those objects with respect to you.

Relating A with Respect to B

Imagine two persons namely A and B running in a playground in different directions. At any instant, they are at rest with respect to themselves but the surroundings are in motion for them. When A observes B, he observes that there is some velocity of B. The velocity of B observed by A or say the velocity of B with respect to A must be different from the absolute velocity of B i.e., the velocity of B with respect to the ground itself. 

The same can be said for the velocity of A with respect to B i.e. when B is observing A. So, for either case, let it be the velocity of B with respect to A. It must be the vector difference of absolute velocity of B and absolute velocity of A. 

Relating A with Respect to B

Mathematical Interpretation of Relative Velocity in Kinematics

Continuing the previous case, let us name the absolute velocity of A to be v_a and the absolute velocity of B to be v_b.

The relative velocity formula of person B relative to person A can be written as

\[{{v}_{ba}}={{v}_{b}}-{{v}_{a}}\]

Similarly, the relative velocity of A with respect to B will be

\[{{v}_{ba}}={{v}_{b}}-{{v}_{a}}\]

Here we note that

\[{{v}_{ba}}=-{{v}_{ab}}\]

And,

\[\left| \left. {{v}_{ab}} \right| \right.=\left| \left. {{v}_{ba}} \right| \right.\]

Relative Velocity Examples

• Imagine yourself in a car running on a highway. Now imagine another car passing your car from the side. You’ll observe that the car is passing by at a reasonably slow speed. But when you are standing on the road, that car will appear to be passing very quickly. The former observation is because you are observing that car relative to you and you have some speed too because you are sitting in a moving car. So, another car appears to be passing relatively slowly as it would be if observed from the ground.

• Imagine yourself on the bank of a river. You need to cross the river and reach another end. Note that the river itself is flowing. Now, you jump in and start swimming in the straight line joining the two ends. But when you reach another end, you will see that you are not on the point directly opposite to the point where you began. 

Your location will be shifted and you will find yourself at some other point. It is because you were trying to follow that straight line and according to you, you only have velocity in that direction. But actually, you have the velocity of swimming and the velocity of the water flow itself. So, the total velocity vector is not going to be in the initial straight line joining two opposite points.

• Suppose you are in a train carriage. All the doors and windows are closed so you can’t see anything outside which also means you can’t state if the train is moving or you are at rest. The train might be moving at a uniform velocity or it might be at rest. The train is always going to be on rest with respect to you. There is no experiment in this world that you can conduct to know your kinematic state. 

All the laws of physics will hold true in that train carriage either it is moving at some velocity or it is at rest. We always comprehend things relative to us and we always see velocities relative to our eyes.

Numerical

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

Solution: Consider that the velocity of the river is towards positive x-axis, that is, \[{{v}_{river}}=4\hat{i}\] and of swimmer is towards positive y-axis, that is ${{v}_{swim}}=3\hat{j}$. The resultant or the actual velocity of the swimmer will be

${{v}_{actual}}={{v}_{river}}+{{v}_{swim}}$

${{v}_{actual}}=4\hat{i}+3\hat{j}$

The magnitude of the actual velocity will be

$\left| {{v}_{actual}} \right|=\sqrt{{{4}^{2}}+{{3}^{2}}}$

$\left| {{v}_{actual}} \right|=5m/s$

The direction of the swimmer will be

$\theta ={{\tan }^{-1}}\left( \frac{3}{4} \right)$

$\theta ={{37}^{0}}$

Hence, the actual velocity of the swimmer and the angle are 5m/s and ${{37}^{0}}$.

Conclusion

The concept of relative velocity is applied to those reference frames that are at motion. The velocity of a moving body from the reference of another moving body is called relative velocity. Relative velocities can be found by vector methods. We can take the components of the velocities towards x and y directions and add them. Then the resultant velocity is the relative velocity. Some examples are a stationary tree seen from inside a moving car that appears to be moving and the drivers of the two cars moving at the same velocity appear at rest to each other.