A rocket mass M is launched vertically from the surface of the earth with an initial speed V. Assuming the radius of the earth to be R and negligible air resistance, the maximum height attained by the rocket above the surface of the earth is

Options

A. `(R)/(((gR)/(2V^(2))-1))`
B. `R((gR)/(2V^(2))-1)`
C. `(R)/(((2gR)/(V^(2))-1))`
D. `R((2gR)/(V^(2))-1)`

Answer

C

Explanation

Kinetic energy given to rocket at the surface of earth = change in potential energy of the rocket in reaching from ground to highest point
`(1)/(2) mv^(2) = (mgh)/(1 + (h)/(R)) , (v^(2))/(2) = (g)/((1)/(h) +(1)/(R)) , (1)/( h) + (1)/(R) = (2g)/( v^(2)) - (1)/(2) , (1)/(h)`
` = (2 gR - v^(2))/(v^(2) R)`
` h = (v^(2) R)/( 2 g R - v^(2)) , h = (R)/(((2gR)/(v^(2))-1))`