1. The problem statement, all variables and given/known data An object is fired upwards with an initial velocity that is less than escape velocity. Show that the maximum height obtained by the object is equal to H=(R*H')/(R-H') where R is the radius of the Earth and H' is the height that the body would obtain if the gravitational field was uniform. 2. Relevant equations Energy conservation equation, equations for gravitational acceleration, gravitational potential energy and gravitational force. 3. The attempt at a solution I have tried obtaining some expressions from the law of energy conservation but I wasn't able to prove the statement in the problem. Firstly I wrote KE-PE=-PE, because at maximum height the body does not have any kinetic energy, and I accounted for the potential energy at the surface of the Earth and I obtained an expression for maximum height as follows: H=-2GM/(Vo^2-2GM/r) which I couldn't from then reduce to the expression in the problem. Furthermore, to try to find maximum height if Earth's gravitational field were to be uniform, I said KE=-PE, (choosing the surface of the Earth as the reference point where initial PE=0, and saying that final PE=-mgh), and from there I obtained the expression for maximum height for uniform gravitational field H'=Vo^2/-g. But I am also thinking that I should have been able to use the Toricelli's kinematics equation V^2=Vo^2+2aΔy, and by using that (which I would be free to use since I am assuming uniform gravitational field and thus constant gravitational acceleration), I would obtain Δy=H'=Vo^2/(-2g) which is different from the expression obtained from the energy conservation law. So if you could please give me some hints on proving the statement in the problem, and also tell me how and why did I end up with different expressions for maximum height for uniform gravitational field by using energy conservation law and Toricelli's kinematics equation. Thanks.