A small block of super dense material has a mass equal to the half of the mass of the earth . It is released from the height `H(H lt lt ` radius of the earth) . Find its speed when its height from the eath surface decreases by `75%`.

Explanation

Since the mass of the block is comparable to the earth , hence the earth will move.
Take (block + earth) as system:
By the momentum conservation
`(M_(e))/(2) v_(1) = M_(e) v_(2) rArr v_(2) = (v_(1))/(2)`
By the energy conservation
`(M_(e))/(2) gH = (1)/(2) (M_(e))/(2) v_(1)^(2) + (1)/(2) M_(e) v_(2)^(2) + (M_(e))/(2) g (H)/(4)`
`(M_(e) gH)/(2) = (M_(e))/(2) g (H)/(4) + (1)/(2) (M_(e))/(2) v_(1)^(2) + (1)/(2) M_(e)((v_(1))/(2))^(2)`
`(3)/(8) M_(e) g H = (3)/(8) M_(e) v_(1)^(2) rArr v_(1) = sqrt(gH)`