A rocket is launched straight up from the surface of the earth. When its altitude is ( 1 3 ) of the radius of the earth, its fuel runs out and therefore it coasts. If the rocket has to escape from the gravitational pull of the earth, the minimum velocity with which it should coast is (Escape velocity on the surface of the earth is (11.2 kms ⁻¹ ).)

Step-by-step solution

Given, escape velocity on the surface of the earth. (v_e= 2 g R_e =11.2 ~km / s )...(i) where, (R ) is the radius of earth. Height of the rocket from the surface of earth, (h= R_e 3 ). If the rocket has to escape from the gravitational pull of the earth from (h= R_ 3 ). Then, the potential energy at the altitude, ( aligned R_e 3 & = kinetic energy G M_e m R_e+h & = 1 2 m v_ e₁ ^2 G M_e R_e+ R_e 3 = 1 2 v_ e₁ ^2 aligned ) ( [v_ e₁ = . ) escape velocity of rocket from ( . R_e 3 ] ) ( ( array rl M_e & = mass of the ea