A very small groove is made in the earth, and a particle of mass `m_(0)` is placed at `(R)/(2)` distance from the centre. Find the escape speed of the particle for that place.

Explanation

Suppose we porjected the particle with speed `v_(e)` so that it just reaches at `(r rarr infty)`
Applying energy conservation
`K_(i)+U_(i)=K_(f)+U_(f)`
`rArr (1)/(2)m_(0)v_(e)^(2)+m_(0)[-(GM_(e))/(2R^(3)){3R^(2)-((R)/(2))^(2)}]=0`
`rArr` Escape energy `= (1)/(2)m_(0)v_(e)^(2)=(11Gm_(0)M_(e))/(8R)`.