- #1

- 31

- 1

I've been learning calculus and physics for the last year. Lets say i have 2 point masses mass one (m1) and mass two (m2). Their respective velocity vectors are constrained to one spatial dimension, with the directional component of their velocity vectors being determined by their respective signs (+,-). So i will refer to their respective velocity vectors from here on as simply being a signed magnitude (+2 m/s, -4m/s). Let the velocity vector of point mass one be V1, the velocity vector of point mass 2 be V2. Their respective momentum vectors are therefore Vn*Mn. Now lets assume that the two point masses will collide, and that when they do collide they will stick together.

So heres my question; is there a way that i can determine the rate of momentum transfer with respect to time P' (or the equal and opposite forces) they are experiencing at each moment after they collide until their velocities equalize? This really boils down to whether or not there is a way to determine the equal and opposite forces these two point masses will experience based on their masses and velocity vectors, assuming that from the point of contact onwards they magically stick together. I initially thought I could set up a pair of coupled linear differential equations and find the eigenvalues of the resultant matrix and solve it that way. It soon became apparent that there was no clear way to determine what the contact force should be based on their velocities, but the only way that the system would show the desired behavior (conservation of momentum, asymtotic velocity equilization, collissions with large mass djfferences taking less time) was if the force experienced by m1, p1' = u(v2-v1), where u is some constant, and p2' = -u(v2-v1), and the only constant u I could find that showed the desired behavior (large forces and quick velocity equilization when you have one massive and one very less massive object) was the reduced mass, m1*m2/(m1+m2), so v1'=u/m1(v2-v1) and v2'=u/m2(v1-v2). The end solution appeared to show the correct behavior, with the trajectories through the phase space moving along scalar multiples determined by the initial conditions of one of the eigenvectors towards scalar constants of the second eigenvector (which is a vector pointing at 45 degree or 225 degrees, meaning that the velocities eventually equalize).

Is any of this right?

Thanks for your time :)