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I am doing a project with the Inelastic Collison equation and was wondering whether you can insert a relativistic mass into it or whether you can only use rest masses. If you can, is there any mathematical way to prove this?

Here is what I am trying to do:

In this equation:

$$V=\frac{M_1 V_1 + M_2 V_2}{M_1 + M_2}$$

I want to replace the masses with their relativistic mass where relativistic mass is the following equation:

$$m_{rel} = \frac{m}{\sqrt{1-\frac{v^2}{c^2}}}$$

Qmechanic
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DragonflyRobotics
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  • please make you question more clear. You have of cause to consider the right addition of velocities. – trula Oct 19 '20 at 18:01
  • It is not clear what you are having touble with, you should add some details about your confusion. Please note that the term/concept "relativistic mass" is extremely misleading and therefore not really used by most physicists. – AlmostClueless Oct 19 '20 at 18:02
  • Why do you want to replace them? Why do you think your first equation would remain valid? Go find out about the relativistic definition of momentum. – Bill N Oct 19 '20 at 18:43
  • Modern treatments of Special Relativity avoid relativistic mass because it can be confusing & misleading. Please see https://physics.stackexchange.com/q/133376/123208 – PM 2Ring Oct 19 '20 at 21:13

1 Answers1

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I want to replace the masses with their relativistic mass where relativistic mass is the following equation:

Yes, you can. But remember that you had better write the relativistic law of momentum conservation. When you do, you will find that it is as if you replace the masses in your classical equation with the corresponding relativistic masses. However, remember that there is not only one gamma factor in your equation but rather you will have three different gammas:

$$p_1+p_2=P\space,$$

where we denote by $p_i$ the momentum of the $i^{th}$ object before the collision, and by $P$ the momentum of the stuck objects, combining as one object through an inelastic collision, after the collision. Therefore, you have:

$$\gamma_{V_1}M_1V_1+\gamma_{V_2}M_2V_2=\gamma_{V}M'V\space.$$

Also remember that, in inelastic collisions, you are not allowed to judge the final mass of the combination to be $M_1+M_2$ because additional energy changes into matter, which consequentially changes the rest mass of the combination ($M'$) to something greater than $M_1+M_2$. Therefore, you need the conservation of the relativistic total energy (before and after are the same as in the previous equation) to help you to find $M'$ via the following equation:

$$\gamma_{V_1}M_1c^2+\gamma_{V_2}M_2c^2=\gamma_{V}M'c^2\space.$$

You need both of the above equations to solve $M'$ and $V$.

Mohammad Javanshiry
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