Question

A ball with a mass of `3` `kg` is rolling at `18` `ms^-1` and elastically collides with a resting ball with a mass of `9` `kg`. What are the post-collision velocities of the balls?

Answer 1

In an elastic collision, both momentum and kinetic energy are conserved. If we call the `3` `kg` ball 1 and the `9` `kg` ball 2, their final velocities are `v_1=-14.4` and `v_2=10.8` `ms^-1`.

Explanation:

Initial momentum:

`p=m_1v_1+m_2v_2=3*18+9*0=54` `kgms^-1`

Initial kinetic energy:

`E_k=1/2m_1v_1^2+1/2m_2v_2^2=1/2*3*18^2+1/2*9*0^2=486` `J`

Because this is an elastic collision, both will be conserved:

Final momentum:

`p=54=m_1v_1+m_2v_2=3v_1+9v_2` - call this `(1)`

Final kinetic energy:

`E_k=486=1/2m_1v_1^2+1/2m_2v_2^2=1/2*3*v_1^2+1/2*9*v_2^2` - call this `(2)`

We now have two equations, `(1)` and `(2)`, in two unknowns.

Let's double `(2)`, just for neatness:

`972=3v_1^2+9v_2^2`

We can find a value for `v_1` by rearranging `(1)` and then substitute that into this revised `(2)` so that we are working with only one variable:

`54=3v_1+9v_2`

`v_1=(54-9v_2)/3`

Then:

`972=3((54-9v_2)/3)^2+9v_2^2=(54-9v_2)^2/3+9v_2^2`

`972=(2916-972v_2+81v_2^2)/3+9v_2^2`

Let's multiply through by 3 for neatness:

`2916=2916-972v^2+81v_2^2+9v_2^2`

Rearranging:

`90v_2^2-972v_2=0`

Solve using the quadratic formula or otherwise, and you get:

`v_2=(972+-sqrt(944784-0))/180=(972+-972)/180`

There are actually two roots, therefore, `0` and `10.8` `ms^-1`.

That is, the second ball, with a mass of `9` `kg` can be either stationary or moving at `10.8` `ms^-1`.

We should solve `(1)` with both of these to find the possible values of `v_1`:

If `v_2=0`, `v_1=54/3=18ms^-1`

But wait: this is just the condition before the collision! The `3` `kg` ball has a velocity of `18` `ms^-1` and the `9` `kg` ball is stationary!

If `v_2=10.8`, `v_1=-14.4` `ms^-1`.

The minus sign indicates that this velocity is in the opposite direction to the original velocity.

The `3` `kg` ball approaches at `18` `ms^-1` then collides and bounces backward at `14.4` `ms^-1`, propelling the `9` `kg` ball forward at `10.8` `ms^-1`.