# Collision using Reduced mass [closed]

While reading a wikipedia article about collision using reduced mass I came across this.

how has it been derived?

• What have you tried so far? Sep 10, 2020 at 19:27
• @Philip I tried writing KE. of a system using reduced mass. which would contain (KE)of com. and 1/2(mu)vrel^2 but I dont know where to introduce the collision factor. Sep 10, 2020 at 19:30
• @shreya you need to format using math Jax. Search math Jax tutorial Nov 16, 2020 at 9:36

Say there are 2 masses with velocities $$u_1,u_2$$. Let's define the relative velocity $$v$$ as $$v = u_1-u_2$$. Let's also define $$(1)\Delta K = \frac{1}{2}\mu (v_{final}^2 - v_{initial}^2)$$ and use the Coefficent of restitution definition $$e = |\frac{v_{final}}{v_{initial}}|$$ where $$v_{initial}$$ and $$v_{final}$$ are the velocities before and after a collision between the two masses, respectively.
If we multiply and divide (1) by $$v_{initial}^2$$, we get: $$\Delta K = \frac{1}{2}\mu v_{initial}^2 [(\frac{v_{final}}{v_{initial}})^2 -1 ]$$ We can now substitute $$e$$ in to finally get the desired expression for the change in kinetic energy: $$\Delta K = \frac{1}{2}\mu v_{initial}^2 [e^2 -1 ]$$