Question

Start with the equations for initial and final momenta and kenetic energies and derive the theoretical equation for the ratio of Kf to Ki

Answer 1

well you need to keep in mind that work done on a body is stored in the form potential energy and that potential energy is further converted into kinetic energy whilst motion.
now let's say there is a body of mass m and initial velocity Vi and final velocity of Vf , change in momentum = mVf-mVi) and relation between kinetic energy and momentum is \[K.E= \frac{ 1 }{ 2 }mV^2= \frac{ 1 }{ 2 }*P*V\]\[(K.E)_{f}= \frac{ 1 }{ 2 }(P _{f})*V _{f}\] and \[(K.E)_{i}=\frac{ 1 }{ 2 }*(P _{i})*Vi\] their ratio is given by \[\frac{ (K.E)_{f} }{ (K.E)_{i} }=\frac{ P _{f} }{ P _{i} }*\frac{ V _{f}}{ Vi }=\frac{ mV _{f} }{ mV _{i} }*\frac{ V _{f} }{ V _{i} }=\frac{ V _{f}^2 }{ V _{i}^2 }\]

Answer 2

hope that helps you