Coefficient of restitution and collisions in two dimensions

How bouncy is a collision?

Between elastic and sticking lies every real collision. The coefficient of restitution measures where:

It runs from $e = 0$ (they stick) to $e = 1$ (elastic). Together with momentum, it fixes the velocities after any head-on collision.

A ball bouncing on a floor

The floor does not move, so each bounce multiplies the ball's speed by $e$. Since $h = \dfrac{v^2}{2g}$, each bounce multiplies the height by $e^2$:

That gives a simple way to measure $e$: drop a ball, and $e = \sqrt{\dfrac{h_1}{h_0}}$.

You should find the heights falling by the same factor each time, quickly for a small $e$, slowly for one close to $1$.

The ballistic pendulum

A bullet is too fast to time with a stopwatch, but a hanging block can measure it.

Use the right law for each step:

1. The hit (very short, the block barely moves): momentum, $mu = (m + M)V$. Kinetic energy is not kept here.
2. The swing (only gravity does work): energy, $\tfrac12(m + M)V^2 = (m + M)gh$.

Collisions in two dimensions

When bodies collide off-centre, they fly off at angles. Momentum is still conserved, but as a vector: keep the $x$ part and the $y$ part separately.

For equal masses in an elastic glancing collision with one at rest, the two paths afterwards are at right angles.

Watch out. Do not add momenta at angles as plain numbers. Split them into parts first.

A worked example

A $12\ \text{g}$ pellet is fired into a $1.2\ \text{kg}$ block on strings, which swings up $7.5\ \text{cm}$.

1. Swing. $V = \sqrt{2gh} = \sqrt{2 \times 9.8\ \text{m/s}^2 \times 0.075\ \text{m}} = 1.21\ \text{m/s}$.
2. Hit. $u = \dfrac{m + M}{m}V = \dfrac{1.212\ \text{kg}}{0.012\ \text{kg}} \times 1.21\ \text{m/s} = 122\ \text{m/s}$.
3. Energy check. The pellet had $\tfrac12 \times 0.012\ \text{kg} \times (122\ \text{m/s})^2 = 89\ \text{J}$; the swing needs only $1.212\ \text{kg} \times 9.8\ \text{m/s}^2 \times 0.075\ \text{m} = 0.89\ \text{J}$. About $99\%$ became heat.

Think first. Name the steps of the motion, and choose momentum or energy for each.

Practise with new numbers

Try it in Python

The program follows a ball through six bounces and adds up the distance it travels.

Remember these

• $e = \dfrac{\text{speed of separation}}{\text{speed of approach}}$, from $0$ to $1$.
• A bouncing ball: $h_n = e^{2n}h_0$.
• Ballistic pendulum: momentum for the hit, energy for the swing.
• In two dimensions, conserve momentum along each axis.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A ball dropped from $1.25\ \text{m}$ bounces to $0.80\ \text{m}$. What is $e$?

2. With $e = 0.60$, what fraction of the drop height does a ball reach after two bounces?

3. In a ballistic pendulum, why can you not use conservation of kinetic energy for the hit?

4. A $2.0\ \text{kg}$ puck moving east at $3.0\ \text{m/s}$ sticks to a $2.0\ \text{kg}$ puck moving north at $3.0\ \text{m/s}$. In what direction do they move?