A particle is projected vertically upwards from the surface of a planet of radius R . The projection speed is given as v = 1 3 v_ e , where v_ e is the escape velocity from the surface of the planet. The maximum height reached by the particle from the surface of the planet is :

Step-by-step solution

The escape velocity from the surface of the planet is v_ e = 2GM R . The initial speed of the particle is v = 1 3 v_ e = 2GM 3R . Applying the principle of conservation of mechanical energy between the surface of the planet and the maximum height h : K_ i + U_ i = K_ f + U_ f 1 2 mv² - GMm R = 0 - GMm R+h Substituting the value of v² : 1 2 m ( 2GM 3R ) - GMm R = - GMm R+h GMm 3R - GMm R = - GMm R+h - 2GMm 3R = - GMm R+h 2 3R = 1 R+h 2(R+h) = 3R 2R + 2h = 3R 2h = R h = R 2 . Answer: R 2