A ball is thrown down a lawn in such a way that it initially slides with a speed `v_(0)` without rolling. It gradually picks up rotational motion. Prove that it will be without sliding, that is, its motion will be pure rolling when its speed falls to `5/7v_(0)`.

Explanation

We can see that the frictional force f, weight mg and normal contact force N pass through the point of contact P. Therefore, their torque about P will be equal to zero.
`rArr` Angular momentum of the sphere remains constant before and after start of pure rolling about P.
Conseving the angular momentum of the sphere at position 1 & 2, about the instantaneous point of contact. We obtain
`mv_(0) r = mvr + ((2)/(5)mr^(2)) omega`
Putting `r omega = v` for pure rolling, we obtain `v = 5 v_(0)//7`.