I have a hard time understanding the variation of mass with velocity, more precisely the proof. In almost every material I've found, the author analyses 2 bodies colliding. The idea of looking at the collision is not hard to grasp and by considering one of the velocities equal zero, you get a nice relation between the masses of the two bodies. You cand thus take the resting mass to be m(adsbygoogle = window.adsbygoogle || []).push({}); _{0}and the other m and congrats, you've done it!

Here is a link to such a proof: http://www.physics-assignment.com/variation-of-mass-with-velocity

Here are the exact points I don't understand:

1) The example starts with two bodies and finishes with a relation describing only one body.

2) We make a mass be m_{0}and say that the other varies with respect to it. But m_{0}is supposed to be the mass of the moving body if it would be at rest. So, does this mean that the bodies used have the same mass, no matter what?

I've also found something different and very easy to understand. From what I heard, this is a proof designed by Einstein himself:

https://proofwiki.org/wiki/Einstein's_Mass-Velocity_Equation

I like this one and I have no problem with its validity. But it seems a bit like a particular case. What I mean is that it starts from a restrictive scenario: "The space ship's trajectory be perpendicular to the comet's trajectory towards the planet, so there is no length contraction parallel to the trajectory of the comet."

Is this second proof really general or not.

A third one I've found in a book that is not printed in English, from what I know. It uses space-time vectors and is very short. But is not well enough explained in the book at one particular point. Maybe two.

I follows as this:

x^{μ}= (ct, x, y, z)

u^{μ}= dx^{μ}(τ)/dτ = (c dt/dτ, dx/dτ, dy/dτ, dz/dτ)

Where τ = Δs/c (the studied body's time)

*Here I have the first lack of understanding with this proof: why use τ (the body's time). What significance has using the ratio of the coordinates measured by you and the time measured by the moving body. After all, in the body's frame, it is not moving with respect to itself.

u^{μ}= dt/dτ (c, dx/dt, dy/dt, dz/dt)= dt/dτ (c, v_{x}, v_{y}, v_{z}) = γ(c, v_{x}, v_{y}, v_{z})

Where γ = 1/√(1-v^{2}/c^{2})

The author says thatv= (v_{x}, v_{y}, v_{z}) is the speed measured by us. So if we multiply by the rest mass m_{0}we get:

p^{μ}= m_{0}γ(c, v_{x}, v_{y}, v_{z})

**Here I really don't get it: why is p^{μ}= m_{0}u^{μ}. If we differentiate x^{μ}by our time t and then multiply by m_{0}we simply make γ disappear, so the real question here is: what's the difference.

***Also, could anyone explain what u^{μ}= γ(c, v_{x}, v_{y}, v_{z}) actually represents?

Note: I don't say that any of those proofs are wrong, I just want to understand them better. I would also be grateful if anyone has an easier proof or some material that explains one of those better.

Note: I would also be grateful for a nice source of problems for both special and general relativity. What I have found so far is usually just easy exercises that ask you to apply Lorenz transforms, which aren't actual problems. I'm looking for more challenging stuff, that involves Minkovski space-time representation, space-time interval and things like those.

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# I Difficulty in understanding the proof of mass-speed relation

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