Question

A ball with a mass of `5 kg` is rolling at `7 m/s` and elastically collides with a resting ball with a mass of `4 kg`. What are the post-collision velocities of the balls?

Answer 1

I found: `0.78 and 7.78 m/s` but check my maths!

Explanation:

We can use conservation of momentum `vecp=mvecv` in one dimension (along `x`). So we get:

`p_("before")=p_("after")`

So:

`(5*7)+(4*0)=(5*v_1)+(4*v_2)`

Being an elastic collision also kinetic energy `K=1/2mv^2` is conserved:

`K_("before")=K_("after")`

So:
`1/2*5*7^2+1/2*4*0^2=1/2*5*v_1^2+1/2*4*v_2^2`

We can use the two equations together and we get:

`{(35=5v_1+4v_2),(245=5v_1^2+4v_2^2):}`

From the first equation:
`v_1=(35-4v_2)/5`
Substitute into the second and get:
`245=cancel(5)(35-4v_2)^2/cancel(25)^5+4v_2^2`
`cancel(1225)=cancel(1225)-280v_2+16v_2^2+20v_2^2`
`36v_2^2-280v_2=0`
`v_2=0m/s` (not) and `v_2=280/36=70/9=7.78m/s` yes.
Corresponding to:
`v_1=7m/s` and `v_1=7/9=0.78m/s`