How can we derive equations of collision without momentum

I'm wondering how we can predict how velocities following a collision of two masses will change post collision without relying on expectations as to what will happen in a particular type of collision.

With two masses $m_1$ and $m_2$ at initial velocities along a one dimensional axis $v_{1i}=v_x\hat{i}$ and $v_{2i}=0\hat{i}$ where $c_1$ and $c_2$ are some constant velocities, we have conservation of linear momentum.

I'm wondering if it is possible to theoretically predict what will happen in various cases of collisions without apriori knowledge of what the properties of these collisions are.

In our case, how can we predict that when $m_1$ hits $m_2$ it will transfer all its momentum to $m_2$? Sure if we already knew that its velocity goes to $0$ we could use the equation of momentum conservation to solve for the final velocity of $m_2$ or vice versa. But if we didn't know that $v_1$ goes to $0$ post collision, how could we get this result theoretically and not empirically?

We can make some assumptions about the deformation behavior of the objects as well as consider the various cases of collisions. I'm interested in this as an extension of what I'm learning in my Physics I lectures.

• With Newton's laws and a detailed understanding of the deformation behavior of both objects. This is a hard problem. – dmckee Nov 2 '16 at 3:12
• Ok. I thought so. I tried hard but had problems. Now I'm asking a more clear version of the question hoping for some direction. – theideasmith Nov 2 '16 at 3:28
• Consider this: Deformation of the object uses up the Kinetic Energy, not the momentum. – MrYellow Nov 2 '16 at 3:43
• We need to derive the post collision velocities without knowing what either of them are experimentally in various situations. – theideasmith Nov 2 '16 at 3:47
• If the collision is between 2 rigid circles or spheres, conservation of energy (or coefficient of restitution) and momentum is enough to predict the outcome. eg Elastic collision between two circles – sammy gerbil Nov 3 '16 at 2:09