A shell is fired vertically from the earth with speed `v_(esc)//N`, where `N` is some number greater than one and `v_(esc)` is escape speed on the earth. Neglecting the rotation of the earth and air resistance, the maximum altitude attained by the shell will be (`R_(E)` is radius of the earth):

Options

A. `(R_(E))/(N^(2)-1)`
B. `(R_(E))/(N^(2))`
C. `(NR_(E))/(N^(2)-1)`
D. `(N^(2)R_(E))/(N^(2)-1)`

Answer

A

Explanation

`-(GMm)/R=-(GMm)/(R+h)+1/2mv^(2)`
`1/2v^(2)=GM(1/(R+h)-1/R)v^(2)=(2GM)/Rh/(R+h)`
`v^(2)=v_(e)^(2)h/(R+h)` given `v=(v_(e))/N`
Substitute `rArr h=R/(1-N^(2))`