A particle is projected vertically upwards from the surface of earth (radius R) with a kinetic energy equal to half of the minimum value needed for it to escape. The height to which it rises above the surface of earth is

Options

A. R
B. 2 R
C. 3 R
D. 4 R

Answer

A

Explanation

Decrease in kinetic energy = increase in PE
`:. (1)/(2)m((v_(e))/(sqrt(2)))^(2)=(mgh)/(1+(h)/(R))or(v_(e)^(2))/(4)=(gh)/(1+(h)/(R))`
or `(2gR)/(4)=(gh)/(1+(h)/(R))or (R)/(2)=(h)/(1+(h)/(R))`
Solving this equation, we get `h=R`
Note : Kinetic energy is half the value required to escape. Therefore, speed is `1//sqrt(2)` times the value required to escape.