How can we derive equations of collision without momentum

Question

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 --- ex-moderator kitten, Nov 02 '16 at 03:12
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 02 '16 at 03:28
Consider this: Deformation of the object uses up the Kinetic Energy, not the momentum. — MrYellow, Nov 02 '16 at 03:43
We need to derive the post collision velocities without knowing what either of them are experimentally in various situations. — theideasmith, Nov 02 '16 at 03:47
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 03 '16 at 02:09

score 0 · Accepted Answer · answered Nov 02 '16 at 07:44

There are three important points:

1- It's important that whether the energy is conserved or not during the collision.

2- A thorough analysis of the collision is almost impossible without knowing the details of interactions between particles, except in very simple cases (see next paragraph).

3- Conservation laws-like momentum or energy- are usually detours to find the solutions of problems in physics. This is partly because when you attempt to solve the problem by solving the differential equations of motion of the system, after performing some sufficient integrations you'll get these conservation laws-and hence conserved quantities are sometimes called integrals of motion-. In some cases (like when the collision occurs in one dimension of space) these conservation laws solely can give us the complete solution of the problem. But when problems get more complicated these conservation laws aren't sufficient to give the solution of problems. As a result in one dimension until the conservation of energy is preserved it doesn't matter that what's the form of the interaction between two particles, whether there is a repulsive coulomb interaction or a hard ball interaction between them, their final velocities will be the same -with similar initial conditions-. But in two or more dimensions one can see that occasionally there is not any other ways than using brute force to solve the equations of motion.

How can we derive equations of collision without momentum

1 Answers1