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

collision of particles how has it been derived?

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    $\begingroup$ What have you tried so far? $\endgroup$
    – Philip
    Sep 10, 2020 at 19:27
  • $\begingroup$ @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. $\endgroup$
    – shreya
    Sep 10, 2020 at 19:30
  • $\begingroup$ @shreya you need to format using math Jax. Search math Jax tutorial $\endgroup$
    – David
    Nov 16, 2020 at 9:36

1 Answer 1

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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: $$ \Delta K = \frac{1}{2}\mu v_{initial}^2 [e^2 -1 ]$$

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