A body is projected vertically upwards from the surface of a planet. It reaches a maximum height equal to the radius of the planet. The ratio of its projection speed to the escape speed from the surface of the planet is

Step-by-step solution

Let M and R be the mass and radius of the planet, respectively. Let v be the projection speed. The initial total mechanical energy of the body at the surface of the planet is E_ i = - GMm R + 1 2 mv^2 At the maximum height h = R , the distance from the center of the planet is r = R + R = 2R . The kinetic energy is zero, so the final total mechanical energy is E_ f = - GMm 2R By conservation of mechanical energy, E_ i = E_ f : - GMm R + 1 2 mv^2 = - GMm 2R 1 2 mv^2 = GMm 2R v = GM R The escape speed from the surface