A space probe is launched from the surface of the Earth with a speed equal to 1.5 times the escape velocity ( V_e ). The residual speed of the probe when it reaches interstellar space (very far away from the Earth) is

Step-by-step solution

Let the mass of the Earth be M , radius be R , and mass of the probe be m . By conservation of mechanical energy between the surface of the Earth and infinity: K_i + U_i = K_f + U_f At the surface, the velocity is v = 1.5 V_e . At infinity, the potential energy is zero and the velocity is v_ . 1 2 m(1.5 V_e)^2 - GMm R = 1 2 mv_ ^2 + 0 We know that the escape velocity V_e = 2GM R , which gives GM R = V_e^2 2 . Substituting this into the energy equation: 1 2 (2.25 V_e^2) - 1 2 V_e^2 = 1 2 v_ ^2 Multiplying the entire