Show that if a body is projected vertically upward from the surface of the earth so as to reach a height `nR` above the surface
(i) the increase in its potential energy is `((n)/(n + 1)) mgR`,
(ii) the velocity with which it must be projected is `sqrt((2ngR)/(n + 1))`, where `r` is the radius of the earth and `m` the mass of body.

Answer

(i). `[(n)/(n+1)]mgR`
(ii). `sqrt(((n)/(m+1))2gR)`

Explanation

`U_(i)=(GMm)/(R(1+n))impliesU_(i)=-(GMm)/(R)`
`DeltaU_(f)-U_(i)=-(GMm)/(R)[(1)/(1+n)-1]`
by applying enegy conservation
`(1)/(2)mv^(2)=(GMm)/(R)[1-(1)/(1+n)],v=sqrt((2GM)/(R)[1-(1)/(1+n)])`