A body with speed v is projected from the surface of the earth(mass M & radius R). Find the maximum distance from the earth that this body reaches before returning back to earth, as a function of the initial speed v, M, R and the gravitational constant G

This question tests the students' understanding on conservation of energy, gravitational potential and algebraic manipulation.The first step is identifying that the principle to use is the conservation of energy:K.E. initial + P.E. intial =K.E. final + P.E. final .When you substitute in the expressions for the energies this becomes: 1/2 m v² -GMm/R = 1/2 m v²_final -GMm/r_final. Another key step in solving it, is recognising that the maximum height occurs at the point where v_final =0. The rest is just rearranging so that you have r in terms of v,G,M,R until you reach: r =2GMR/(2GM-Rv²). From this expression, a lot of useful information can be gathered, for example one can derive the escape velocity of a body from earth