A particle of mass m is projected with a velocity v = k V e   ( k < 1 ) from the surface of the earth ( V e = escape velocity). The maximum height above the surface reached by the particle is___

Step-by-step solution

v = k V e ( k < 1 ) 1 2 m v 2 - G M m R = 0 - G M m R + h 1 2 m k 2 × 2 G M R - G M m R = - G M m R + h           ∵ V e = 2 G M R G M m k 2 R = - G M m 1 R + h - 1 R k 2 R = h R ( R + h ) ⇒ 1 + R h = 1 k 2 ⇒ R h = 1 k 2 - 1 ⇒ h = R K 2 1 - k 2