# Relative Velocity After Collision: What, How To Find, Problems

In relativity, the velocity of an object relative to the position of another object in another frame of reference is a relative velocity.

The law of conservation of energy and momentum is followed by the particles undergoing elastic or an inelastic collision. The relative velocity after a collision is the difference between the final velocity of the object from its initial velocity.

## What is relative velocity after collision?

Upon collision, the kinetic energy of the incoming objects is exchanged due to action and reaction forces.

The relative velocity is a difference in the final velocity of the object after a collision, minus the initial velocity before the collision. The relative velocity of the two objects approaching each other before a collision is equal to the relative velocity of the object after a collision if the collision is elastic.

If the particle A having a mass of m is moving with the velocities u from infinity towards a stationary particle B, and after a collision, it deflects away with the final velocity v then the relative velocity of the particle is VR=v- u.

## How to calculate the velocity after collision?

The object moves with some velocity after collision with the kinetic energy that remained or gained on collision.

The final velocity of the object can be calculated based on the fact that the momentum of the particle is always conserved before and after a collision. The total energy is also conserved in a process. Thus by equating the total momentum before and after a collision, we can find the final velocity of each particle.

Consider a particle B at rest and particle A is approaching particle B from the infinity with velocity ua. Particle A collides with particle B and transfers a part of its kinetic energy to particle B.

Upon transmission of energy, particle B starts propagating with the velocity vb, and particle A upon colliding exerts a repulsive force due to which its kinetic energy will be slightly reduced and propagate with the velocity va.

According to the law of conservation of momentum, the sum of the momentum of particles A and B before collision will be equal to the sum of the momentum of both the particles after a collision. Hence, we can write an equation depicting the same as below:

From this equation, we can write an equation for the velocity of particle B as:

Based on the law of conservation of energy, the sum of kinetic energy of particle A and B before and after collision remain the same. We can write the equation for the law of conservation of energy from particle A and B as:

Squaring and adding the equation (1) and (2), we have:

Rearranging this equation, we get:

The above equation is a linear quadratic equation, and can be solved to find the values of the variable va.

After calculating the final velocity of particle A, the final velocity of particle B can be calculated using the equation (3) or (4).

## What is relative speed of separation?

The relative speed of separation of the particles is the same as before the collision.

If the collision is elastic, then the sum of the velocities of the particles before colliding and the sum of the final velocities both are equal, because the linear momentum of the particle is conserved.

The speed of separation is the speed gained by the particle after overcoming the collision and the work done due to collision is the change in the potential energy of the particle. There is a variation in the kinetic and potential energy of the particle due to which its velocity is changed.

## How to calculate relative speed of separation?

The speed is measured in a given time from one frame of reference, while the relative speed is the speed measured in another frame of reference.

The relative speed of separation is the change in the final velocity of the objects colliding with each other and the initial velocity of the object before colliding.

Upon separation of the particles after the collision, they will start propagating with some velocities. Based on the direction of propagation of the particles, we can calculate the relative speed of separation of the particles.

## Relative Velocity after Collision Examples

Consider two people doing a boat ride on a lake. By mistakenly, one boat rider in boat A approaches another boat rider in boat B and dashes over it. After this collision, boat A comes to a sudden rest and boat B sways a little away from its initial state. The relative velocity of boat A with respect to boat B is equal to the velocity with which it is moving after a collision.

Let us take a second example of marbles. Upon hitting the marble towards the other marble which is stationary, the marble at the stationary position will gain the kinetic energy of the approaching marble and accelerate in the direction in which the marble is thrown from the hand.

The marble which was in the hand will change its direction after colliding and travelling in another direction. In this scenario, the velocity of this marble will either be lost or gained after the collision. The relative velocity of the marble is the difference between its initial velocity to its final velocity.

## Relative Velocity after Collision Problems

### What is the relative velocity of the billiard balls of mass 180 grams if the initial velocity of one ball is 3m/s and another ball was stationary?

Given: The initial velocity of ball1 is, u1 =3 m/s

The initial velocity of ball2 is, u2 =0

The mass of ball is, m=180 grams= 0.18 kg

The collision is elastic, hence according to the law of conservation of mass,

Hence, v=u1/2

Substituting the values in this equation, we get:

v=3 m/s/2=1.5 m/s

The final velocity of the both the balls is 1.5 m/s.

The relative velocity of ball1 is,

The relative velocity of ball2 is,

Hence, the relative velocity of the ball1 is 1.5m/s while the relative velocity of ball2 is -1.5 m/s after a collision.

### The object is moving towards the stationary object with a velocity of 580 m/s and reflects away from the object with a velocity of 485m/s. What is the relative velocity of the object after collision and w.r.t. the stationary object?

Given: The initial velocity of the object is, u=580 m/s.

The final velocity of the object is, v=485 m/s.

The initial and the final velocity of the stationary object is, s=0.

The relative speed of the object after the collision can be calculated by using a formula,

Substituting the values in this equation, we get:

The relative velocity of the object with respect to the stationary object is,

Hence, the relative velocity of the object after the collision is 95 m/s, and the relative velocity of the object with respect to the velocity of the object after the collision is 485 m/s only.

### What is the relative velocity of the stone moving with the velocity of 6m/s after releasing from the slingshot if it fell on the surface of the water and was drawn with a speed of 0.3m/s?

Given: the initial speed of stone is, u=6m/s

The final speed of stone is, v=0.3m/s

Hence, the relative speed of a stone is,

The relative velocity of a stone is -5.7m/s. The acceleration of a stone is in a negative direction about an axis.

#### Conclusion

The relative velocity after the collision is the change in the velocity of the objects after undergoing collision. The change in velocity is due to the transfer of momentum and energy when the particles collide with one another. It follows the law of conservation of momentum and energy.

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