Elastic Collisions

Definition: 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 tranformed permanently through work or deformation of the objects. During the collision the energy is going to be transferred (for example as a ball compresses) but will be recovered during the elastic response of the system (for example the ball then expanding again).

Before the collision

The figure below shows two balls rolling toward each other, about to collide:

Two balls before they collide.

We have calculated the total initial momentum previously, now 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 ball 2 has a kinetic energy which we call ${\text{KE}}_{i2}$, the total kinetic energy before the collision is:

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

After the collision

The figure below shows two balls after they have collided:

Two balls after they collide.

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

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

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{align*} \vec{p}_{Ti}& = \vec{p}_{Tf}\\ \vec{p}_{i1}+\vec{p}_{i2}& = \vec{p}_{f1}+\vec{p}_{f2}\\ & \mathbf{\text{and}} \\ {\text{KE}}_{Ti}& = {\text{KE}}_{Tf}\\ {\text{KE}}_{i1}+{\text{KE}}_{i2}& = {\text{KE}}_{f1}+{\text{KE}}_{f2} \end{align*}

Example: An Elastic Collision

Question

Consider a collision between two pool balls. Ball 1 is at rest and ball 2 is moving towards it with a speed of $\text{2}$ $\text{m·s$^{-1}$}$. The mass of each ball is $\text{0.3}$ $\text{kg}$. 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?

Step 1: Choose a frame of reference

Choose to the right as positive and we assume that ball 2 is moving towards the left approaching ball 1.

Step 2: Determine what is given and what is needed

Mass of ball 1, ${m}_{1}=0,3~\text{kg}$.
Mass of ball 2, ${m}_{2}=0,3~\text{kg}$.
Initial velocity of ball 1, $\vec{v}_{i1}=\text{0}\text{ m·s$^{-1}$}$.
Initial velocity of ball 2, $\vec{v}_{i2}=\text{2}\text{ m·s$^{-1}$}$ to the left.
Final velocity of ball 2, $\vec{v}_{f2}=\text{0}\text{ m·s$^{-1}$}$.
The collision is elastic.

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

momentum is conserved, ${m}_{1}\vec{v}_{i1}+{m}_{2}\vec{v}_{i2}={m}_{1}\vec{v}_{f1}+{m}_{2}\vec{v}_{f2}$
energy is conserved, $\frac{1}{2}\left({m}_{1}{v}_{i1}^{2}+{m}_{2}{v}_{i2}^{2}\right)=\frac{1}{2}\left({m}_{1}{v}_{f1}^{2}+{m}_{2}{v}_{f2}^{2}\right)$

Step 3: Draw a rough sketch of the situation

Step 4: Solve the problem

Momentum is conserved in all collisions so it makes sense to begin with momentum. Therefore:

\begin{align*} \vec{p}_{Ti}& = \vec{p}_{Tf}\\ {m}_{1}\vec{v}_{i1}+{m}_{2}\vec{v}_{i2}& = {m}_{1}\vec{v}_{f1}+{m}_{2}\vec{v}_{f2}\\ \left(0,3\right)\left(0\right)+\left(0,3\right)\left(-2\right)& = \left(0,3\right)\vec{v}_{f1}+0\\ \vec{v}_{f1}& = -\text{2.00}\text{ m·s$^{-1}$} \end{align*}

Step 5: Quote your final answer

The final velocity of ball 1 is $\text{2}$ $\text{m·s$^{-1}$}$ to the left.

Elastic Collisions