A particle is projected vertically upwards from the surface of the earth with a speed equal to one-half of the escape velocity. What is the maximum height attained by the particle? (Let R be the radius of the earth)

Step-by-step solution

Let the mass of the earth be M and its radius be R . The escape velocity from the surface of the earth is given by v_e = 2GM R . The initial speed of the particle is v = v_e 2 = 1 2 2GM R . By the principle of conservation of mechanical energy: K_i + U_i = K_f + U_f 1 2 mv^2 - GMm R = 0 - GMm R+h Substituting the value of v : 1 2 m ( 1 4 2GM R ) - GMm R = - GMm R+h GMm 4R - GMm R = - GMm R+h - 3GMm 4R = - GMm R+h 3 4R = 1 R+h 3(R+h) = 4R 3R + 3h = 4R 3h = R h = R 3 . Answer: R 3