Law of Conservation of Momentum: $$m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$$
Kinetic Energy Ratio:
$$\frac{\frac{1}{2}m_1 u_1^2 + \frac{1}{2} m_2u_2^2}{\frac{1}{2}m_1v_1^2 + \frac{1}{2} m_2v_2^2}$$
where $m_1$ and $m_2$ are the masses of the two objects, $u_1$ and $u_2$ are initial velocities, and $v_1$ and $v_2$ are the final velocities
Utilizing conservation of momentum and Total kinetic energy of a system, is there any way to show some sort of proportional relation between KE Ratio, and the mass?
NB: Please try to keep the mathematics fairly simple, since I will be using this as part of a hypothesis to confirm a practical high school investigation. Also note that this is a one dimensional collision between 2 objects.
Best Answer
A good way to attack this sort of question is to choose your frame of reference to simplify it. In this case, choose a reference frame so the initial momentum of the whole system is zero. Then you have $$m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2 = 0.$$
Now you can start eliminating things from the second equation. For example you could eliminate $m_2$. Then, the $m_1$'s cancel out and you are left with a relation between the initial and final velocities.
You don't say exactly what you want to do, and in any case this isn't a "do my homework for me" site, but that should give you some ideas about how to get started.
There will be a constant velocity difference between this reference frame and the "lab frame" of course, but you can subtract that at the start and add it back at the end. In any case, in Newtonian mechanics the "kinetic energy of the system" is a different number depending what reference frame you use to measure it, so you probably want to remove that complication from your investigation anyway.
The physics doesn't depend what reference frame you use, but to get a consistent measure of the energy of the system as a number, you need to introduce special relativity, and that is beyond what you are expected to know at high school level!