Assume that there is a tunnel in the shape of a circular arc through the earth. Wall of the tunnel is smooth. A ball of mass m is projected into the tunnel at A with speed v. The all comes out of the tunnel at B and escapes out of the gravity of the earth. Mass and radius of the earth are M and R respectively and radius of the circle shaped tunnel is also . Find minimum possible value of v (call it `v_(0)`) If the ball is projected into the tunnel with speed `v_(0)`, calculate the normal force applied by the tunnel wall on the ball when it is closest to the centre of the earth. It is given that the closest distance between the ball and the centre of the earth is `(R)/(2)`

Answer

`(a) v_(0) =sqrt((2GM)/R)`
(b) `27/4 (GMm)/(R^(2))`