A projectile is fired from the surface of earth of radius `R` with a velocity `k upsilon_(e)` (where `upsilon_(e)` is the escape velocity from surface of earth and `k lt 1)`. Neglecting air resistance, the maximum height of rise from centre of earth is

Options

A. `(R )/(k^(2) - 1)`
B. `k^(2) R`
C. `(R )/(1 - k^(2))`
D. `kR`

Answer

C

Explanation

`K_(i) + U_(f) = K_(f) + U_(f)`
`(1)/(2) m (k nu_(e))^(2) - (GMm)/(R ) = 0 - (GMm)/(r )`
`:. (1)/(2)k^(2) (2gR)- (GM)/(R ) = - (GM)/(r)`
or `(GM)/(r ) (1 - k^(2)) = (GM)/(r )` (as `gR = (GM)/(R ))`
So, `r = (R )/(1 - k^(2))`