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This post is in reference to a previous homework post, found here:

https://www.physicsforums.com/threads/show-that-f-gamma-3-ma.338744/

That thread is closed to further replies. Probably because it's nearly 10 years old.

That thread is about deriving relativistic force from the derivative of relativistic momentum specifically when the force is parallel with the velocity. The OP must prove that F = (d/dt) (γmv) = γ

^{3}ma.

I followed that thread fine and I see where the answer came from. However, when I first attempted the problem (before I found my way here), I was assuming that the velocity of the particle was not necessarily the same as the velocity of the inertial frame buried inside of the lorentz factor γ . Feedback from the community suggested that they are necessarily the same and, indeed, the proof would be impossible if they were not the same.

Why are they the same?

I thought perhaps they are the same because we assume that the velocity vector is the reference frame. In other words, we are traveling on the back of the particle moving at velocity v. Is there a simpler way to think about the problem?