Question

Two identical balls are in contact on a table . A third identical ball strike them symmeritically and come to rest after impact . The coefficient of restitution is? Thanks

Answer 1

Let `m` be mass of all three identical balls. Let the two stationary balls be positioned on the `y`-axis.

Let `uhatx` be velocity of striking ball.
As such initial momentum`=m uhatx`.

At the time of impact, which is given as symmetrical, the three centers of balls make an equilateral triangle.

By symmetry, after impact, both stationary balls will move with equal velocity along their respectively lines of impact which make an angle of `30^@` with `x`-axis.
Let `vecv_1and vecv_2` be velocities of two balls respectively, where `|vecv_1|and |vecv_2|=v`.

Due to Law of Conservation of momentum, the `y` component of both balls will be equal and opposite to each other. Therefore, equating initial momentum with `x` components of momentum of both balls we get
`m u=2mvcos30^@`
`=>u=2vsqrt3/2`
`=>u=sqrt3v` ......(1)

From Newton's law of Restitution we know that coefficient of restitution `e` is given by the expression

`e="relative velocity after the collision"/"relative velocity before the collision"` .....(2)

In the instant case we use velocity component of striking ball along the line of impact with one of balls
`e=(v-0)/(ucos30^@-0)`
`=>e=(v)/(usqrt3/2)`
Using (1) we get
`e=(v)/(sqrt3vsqrt3/2)`
`=>e=2/3`