Match List-I with List-II. List-I (Velocity of projection from Earth's surface) List-II (Maximum height reached) (A) V_e 2 (I) R 3 (B) V_e 2 (II) R (C) 3 V_e 2 (III) 2R (D) 2 3 V_e (IV) 3R ( V_e is the escape velocity and R is the radius of the Earth) Choose the correct answer from the options given below:

Step-by-step solution

Let the velocity of projection be v = k V_e . By conservation of mechanical energy: 1 2 mv^2 - GMm R = - GMm R+h Since V_e = 2GM R , we have v^2 = k^2 2GM R . 1 2 m (k^2 2GM R ) - GMm R = - GMm R+h GMm R (k^2 - 1) = - GMm R+h R+h = R 1-k^2 h = Rk^2 1-k^2 (A) For v = V_e 2 , k = 1 2 h = R(1/2) 1 - 1/2 = R (B) For v = V_e 2 , k = 1 2 h = R(1/4) 1 - 1/4 = R 3 (C) For v = 3 V_e 2 , k = 3 2 h = R(3/4) 1 - 3/4 = 3R (D) For v = 2 3 V_e , k = 2 3 h = R(2/3) 1 - 2/3 = 2R Thus, the correct matching is (A)-(II), (B)-(I), (C)-