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 project the particle with speed `v_(e)`, so that it just reaches infinity `(r to oo)`.
Applying energy conservation principle
`K_(i)+U_(i)=K_(r)+U_(f)`
`(1)/(2)m_(0)v_(e)^(2)+m_(0)[-(GM_(e))/(2R^(3))]{3R^(2)-(( R )/(2))^(2)}=0`

`impliesV_(e)=sqrt((11"GM"_(e))/(4R))`