A space probe is to be launched from the surface of a planet such that it retains a speed equal to the escape velocity ( v_e ) of the planet when it reaches interstellar space (at an infinite distance). What must be the initial projection speed of the probe from the surface of the planet?

Step-by-step solution

By the principle of conservation of mechanical energy, the total energy at the surface of the planet equals the total energy at infinity. K_i + U_i = K_f + U_f 1 2 mv_p^2 - GMm R = 1 2 mv_ ^2 + 0 Since the escape velocity from the surface is v_e = 2GM R , we can write GMm R = 1 2 mv_e^2 . Substituting this into the energy equation: 1 2 mv_p^2 - 1 2 mv_e^2 = 1 2 mv_ ^2 v_p^2 - v_e^2 = v_ ^2 It is given that the desired velocity at infinity is v_ = v_e . Substituting this value: v_p^2 - v_e^2 = v_e^2 v_p^2 = 2v_e^2 v