Question

A ball with a mass of `2 kg` is rolling at `2 m/s` and elastically collides with a resting ball with a mass of `9 kg`. What are the post-collision velocities of the balls?

Answer 1

The initially moving ball would move `1.273 m/s` to the left while the ball initially at rest would move `0.72m/s` to the right.

Explanation:

(NOTE: velocities with apostrophes denotes the arbitrary object's final velocity )

Consider the total momentum of the system.

`mv_1+mv_2 = mv'_1+mv'_2`
Note that final velocities are positive because I assumed them to move to the right after the impact.

`mv_1+mcancel(v_2) = mv'_1+mv'_2`
`(2)(2) = (2)(v'_1)+(9)(v'_2 )`

Moreover, the problem states that the impact involves an elastic collision. Therefore, the coefficient of restitution of the system, `e`, is 1.
`e = 1=(v'_2-v'_1)/(v_1-v_2) = (v'_2-v'_1)/(2-0)`

`therefore,2 =v'_2-v'_1`

To solve the post-velocities solve the system of equations:
`(1) : 9v'_2 +2v'_1=4`
`(2) : v'_2-v'_1=2`

Where `v'_1= -1.273`
and `v'_2 = +0.72`

Since `v'_1` is negative, the assumed direction of its final velocity is the opposite.