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A small ball on a frictionless horizonta...

A small ball on a frictionless horizontal surface moves towards right with velocity v. It collides with the wall and returns back and continues to and fro motion. If the average speed for first to and fro motion of the ball is `((2)/(3))`v, then the coefficient of restitution of impact is

Text Solution

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The ball moves from A to B with a speed `v_(1)=v` and returns from B to `^^` with a speed `v_(2)` say
Since, `vec(v)_(2)=-e vec(v)_(1)`
`rArr v_(2)(-hat(i))=-ev_(1)hat(i)`
`rArr v_(2)=ev_(1)=ev " " (because v_(1)=v)`
Therefore

`vec(v)=(AB+BA)/((AB)/(v_(1))+(BA)/(v_(2)))=(2v_(1)v_(2))/(v_(1)+v_(2))`
Substituting respective values, we get
`v_(1)=v` & `v_(2)=ev`
we obtain,
`vec(v)=(2v(ev))/(v+ev)=(2e v)/(1+e)rArr (vec(v))/(v)=(2e)/(1+e) " "` ...(1)
Since, it is given that `vec(v)=(2//3)v`
`rArr (vec(v))/(v)=(2)/(3) " "` ...(2)
Equating (1) and (2), we obtain,
`rArr e = 0.5`.
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