# How does the mass and velocity affect the elasticity of a collision?

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.

• Trying to use math Jax, but the code keeps failing, sorry if equations are hard to read – ruby duby Aug 29 '17 at 2:50
• MathJax'd it for you. Click on the edit to see the plaintext. – spaceisdarkgreen Aug 29 '17 at 3:10

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.