There is a crater of depth R 100 on the surface of the moon, where R is the radius of the moon. A ball is projected vertically upwards from the crater with a velocity which is equal to the escape velocity from the surface of the moon. If the maximum height attained by the particle is close to n R , where n is an integer, then find the value of n ?

Step-by-step solution

Speed of particle at A , v A = v = 2 G M R At point B , v B = 0 By conservation of energy, decrease in KE = increase in GPE ⇒ 1 2 Mv Λ 2 = U B - U A ∴ 1 2 M · 2 GM R = M V B - V A ∴ G M R = - G M R + h - - G M 2 R 3 - R - R 100 2 R 2 ∴ 1 R = - 1 R + h + 3 2 R - 1 2 99 100 2 · 1 R ∴ 1 R = - 1 R + h + 3 2 R - 1 2 0 . 98 R ∴ 1 R + h = 1 2 R [ 1 - 0 . 98 ] ∴ 2 R = 0 . 02 ( R + h ) ∴ h = 1 . 98 0 . 02 ∴ h = 99 R