# Inelastic Collision

An inelastic collision is a collision in which both bodies stick together and move together after the collision. Momentum remains conserved and kinetic energy initial is always greater than the kinetic energy final for the whole system. These three points will always be true in a perfectly inelastic collision. The momentum of the total system remains conserved not only the momentum of a single particle. As kinetic energy initial is always greater than kinetic energy final so some energy is lost.

Inelastic Collision

## Types of Collision

A collision can be divided into several categories. Three main categories are elastic collision, partially inelastic collision and perfectly elastic collision.

i) Elastic collision: Elastic collision is a collision in which the kinetic energy of two bodies remains the same after and before the collision. Ping pong balls, billiards all are a real-life example of elastic collision.

ii) Perfectly inelastic collision: Perfectly inelastic collision is a collision in which the maximum amount of kinetic energy of a system is lost. The kinetic energy of the system is lost by bonding the two bodies together. The real-life example of perfectly inelastic collision is when a mudball is thrown against a wall and the ball is stick to the wall.

iii) Partially inelastic collision: Partially inelastic collision is a collision in which the object does not stick together.

### Perfectly Inelastic Collision:

Let a particle of mass $$m_{1}$$ with velocity $$u_{1}$$ and a particle of mass $$m_{2}$$ with velocity $$u_{2}$$ collide with each and move together with a velocity of $$V$$ combined. Now we can apply conservation of momentum of the system where initial momentum is equal to final momentum.

Hence, $$m_{1}u_{1}+m_{2}u_{2}=(m_{1}+m_{2})V$$

This is the equation of a perfectly inelastic collision. Now talking about the $$e$$. $$e$$ is the coefficient of restitution. $$e$$ is the ratio between the velocity of separation and velocity of approach. We can write $$e$$ as the ratio between relative velocity before the collision and relative velocity after collision.

Hence coefficient of restitution $$e=\frac{v_{2}-v_{1}}{u_{1}-u_{2}}$$

Now if $$e$$is equal to one then we can call the collision as a perfectly elastic collision. As in perfectly elastic collision velocity of separation and velocity of approach is similar. Now in case of perfectly inelastic collision, relative velocity after collision or velocity of separation is zero. So the value of the coefficient of restitution is zero in case of a perfectly inelastic collision. So $$e$$ can derive a collision is which type of collision. If the value of $$e$$ is in between zero to one then the collision is a partially inelastic collision.

### Momentum is conserved but the kinetic energy is lost in inelastic collision:

We know generally momentum and energy are always conserved. The real difference is the fact that momentum is a vector and energy is a scalar. Imagine a low energy ball is moving to the right. Now the ball is consists of molecules that have some energy and momentum associated with them. As momentum is a vector so the momentum of this ball is the sum of all the momentum vectors of each molecule. Now after a small inelastic collision, we can see that each molecule has different momentum and energy but the sum of all their momentum is still same and pointing towards the right. So the momentum is conserved.

Since energy is a scalar so if we increase the kinetic energy of each molecule by collision the total energy of the system also increases. Hence we can say in an inelastic collision momentum is conserved but the kinetic energy is lost. The kinetic energy is transformed into heat or some other energy.

Q.1 The clash of two cars is elastic or not. Give an explanation.

Answer: So when the two cars clash the kinetic energy is not maintained. The kinetic energy transforms into heat, sound, and the deformation of the objects. So in an elastic collision, the kinetic energy remains the same and after the collision, the two objects bounce back to their initial position. So the clash of two cars is not elastic collision. In the case of inelastic collision, the kinetic energy is lost. Hence the clash of two cars is an inelastic collision.

Q.2: Explain why there is the loss of kinetic energy in an inelastic collision.

Answer: In the case of an inelastic collision there is the loss of kinetic energy but the momentum is conserved. As momentum is a vector so the momentum after collision remains the same but kinetic energy is a scalar hence it transforms into some other energy such as sound, heat, or other forms of energy.

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