Two stones are thrown vertically upwards simultaneously with their initial velocities u1 and u2 respectively. Prove that the heights reached by them would be in the ratio of u_1^2 : u_2^2 . (Assume upward acceleration is -g and downward acceleration to be + g).

We know the upward motion, u² = u² -2gh

Or  h = u^2+v^2\over2g

But at the highest point,  v = 0

Therefore, h = u^2\over2g

For first ball,  h1 = u_1^2\over2g

and for second ball,  h2 = u_2^2\over2g

Thus, {h_1\over h_2}={{u_1^2\over2g}\over {u_2^2\over2g} }={u_1^2\over u_2^2}

Or, h1 :h2 = u_1^2 : u_2^2