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

A particle is projected vertically upwards from the surface of earth (

1.	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
Answer» R 2 R 3 R 4 R Solution :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.