# 6.2 Types of collisions

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## Types of collisions

Two types of collisions are of interest:

• elastic collisions
• inelastic collisions

In both types of collision, total momentum is always conserved. Kinetic energy is conserved for elastic collisions, but not for inelastic collisions.

## Elastic collisions

Elastic Collisions

An elastic collision is a collision where total momentum and total kinetic energy are both conserved.

This means that in an elastic collision the total momentum and the total kinetic energy before the collision is the same as after the collision. For these kinds of collisions, the kinetic energy is not changed into another type of energy.

## Before the collision

Before the balls collide, the total momentum of the system is equal to all the individual momenta added together. Ball 1 has a momentum which we call ${p}_{i1}$ and ball 2 has a momentum which we call ${p}_{i2}$ , it means the total momentum before the collision is:

${p}_{i}={p}_{i1}+{p}_{i2}$

We calculate the total kinetic energy of the system in the same way. Ball 1 has a kinetic energy which we call $K{E}_{i1}$ and the ball 2 has a kinetic energy which we call KE ${}_{i2}$ , it means that the total kinetic energy before the collision is:

$K{E}_{i}=K{E}_{i1}+K{E}_{i2}$

## After the collision

[link] shows two balls after they have collided:

After the balls collide and bounce off each other, they have new momenta and new kinetic energies. Like before, the total momentum of the system is equal to all the individual momenta added together. Ball 1 now has a momentum which we call ${p}_{f1}$ and ball 2 now has a momentum which we call ${p}_{f2}$ , it means the total momentum after the collision is

${p}_{f}={p}_{f1}+{p}_{f2}$

Ball 1 now has a kinetic energy which we call $K{E}_{f1}$ and ball 2 now has a kinetic energy which we call $K{E}_{f2}$ , it means that the total kinetic energy after the collision is:

$K{E}_{f}=K{E}_{f1}+K{E}_{f2}$

Since this is an elastic collision, the total momentum before the collision equals the total momentum after the collision and the total kinetic energy before the collision equals the total kinetic energy after the collision. Therefore:

$\begin{array}{ccc}\hfill \mathrm{Initial}& & \mathrm{Final}\hfill \\ \hfill {p}_{i}& =& {p}_{f}\hfill \\ \hfill {p}_{i1}+{p}_{i2}& =& {p}_{f1}+{p}_{f2}\hfill \\ \hfill & \mathbf{and}& \\ \hfill K{E}_{i}& =& K{E}_{f}\hfill \\ \hfill K{E}_{i1}+K{E}_{i2}& =& K{E}_{f1}+K{E}_{f2}\hfill \end{array}$

Consider a collision between two pool balls. Ball 1 is at rest and ball 2 is moving towards it with a speed of 2 m $·$ s ${}^{-1}$ . The mass of each ball is 0.3 kg. After the balls collide elastically , ball 2 comes to an immediate stop and ball 1 moves off. What is the final velocity of ball 1?

1. We are given:

• mass of ball 1, ${m}_{1}$ = 0.3 kg
• mass of ball 2, ${m}_{2}$ = 0.3 kg
• initial velocity of ball 1, ${v}_{i1}$ = 0 m $·$ s ${}^{-1}$
• initial velocity of ball 2, ${v}_{i2}$ = 2 m $·$ s ${}^{-1}$
• final velocity of ball 2, ${v}_{f2}$ = 0 m $·$ s ${}^{-1}$
• the collision is elastic

All quantities are in SI units. We are required to determine the final velocity of ball 1, ${v}_{f1}$ . Since the collision is elastic, we know that

• momentum is conserved, ${m}_{1}{v}_{i1}+{m}_{2}{v}_{i2}={m}_{1}{v}_{f1}+{m}_{2}{v}_{f2}$
• energy is conserved, $\frac{1}{2}\left({m}_{1}{v}_{i1}^{2}+{m}_{2}{v}_{i2}^{2}\right)=\frac{1}{2}\left({m}_{1}{v}_{f1}^{2}+{m}_{2}{v}_{f2}^{2}\right)$
2. Choose to the right as positive.

3. Momentum is conserved. Therefore:

$\begin{array}{ccc}\hfill {p}_{i}& =& {p}_{f}\hfill \\ \hfill {m}_{1}{v}_{i1}+{m}_{2}{v}_{i2}& =& {m}_{1}{v}_{f1}+{m}_{2}{v}_{f2}\hfill \\ \hfill \left(0,3\right)\left(0\right)+\left(0,3\right)\left(2\right)& =& \left(0,3\right){v}_{f1}+0\hfill \\ \hfill {v}_{f1}& =& 2\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-1}\hfill \end{array}$
4. The final velocity of ball 1 is 2 m $·$ s ${}^{-1}$ to the right.