A particle is fired vertically upward with a speed of 15 km/s . Determine the speed with which it will travel in interstellar space, assuming only the earth's gravitational field is present.

Step-by-step solution

By conservation of mechanical energy, the total energy at the surface of the earth is equal to the total energy in interstellar space (at infinity). 1 2 mv_i^2 - GMm R = 1 2 mv_f^2 + 0 The escape velocity v_e from the surface of the earth is given by v_e = 2GM R 11.2 km/s . Thus, GMm R = 1 2 mv_e^2 . Substituting this into the energy equation gives: 1 2 mv_i^2 - 1 2 mv_e^2 = 1 2 mv_f^2 v_f = v_i^2 - v_e^2 Given v_i = 15 km/s and v_e = 11.2 km/s , we get: v_f = 15^2 - 11.2^2 v_f = 225 - 125.44 = 99.56 v_f 10 km/s