Deriving Relativistic Momentum: A Guide

[soothsayer]

I'm having some problems trying to figure out how to derive relativistic momentum. The way it was explained to me, classically, p=mv=m(dx/dt),but dx/dt is measured differently in different reference frames. So, if you look at time dilation, t=ɣt' where ɣ= 1/(1-(v/c)^2)^1/2 and t' is time in the moving inertial reference frame. So, dt/dt' = ɣ. m(dx/dt') = m(dx/dt)(dt/dt') = ɣmv. So, p=ɣmv, which is the given formula for special relativistic momentum (though usually u is used instead of v to distinguish velocities of frames and objects). I get all of this.

My question is, don't we have to find p=m(dx'/dt')? if dx/dx' = ɣ and dt/dt' = ɣ, then
m(dx'/dt') = m(dx/dt)(dt/dt')(dx'/dx) = mv(ɣ/ɣ) = mv.
Or, basically, x'=x/ɣ and t'=t/ɣ, so dx/dt = dx'/dt'. Length contracts at the same rate time dilates (dictated by value of ɣ). So where does p=ɣmv come from?

 

[bcrowell]

I'm having some problems trying to figure out how to derive relativistic momentum. The way it was explained to me, classically, p=mv=m(dx/dt),but dx/dt is measured differently in different reference frames. So, if you look at time dilation, t=ɣt' where ɣ= 1/(1-(v/c)^2)^1/2 and t' is time in the moving inertial reference frame. So, dt/dt' = ɣ. m(dx/dt') = m(dx/dt)(dt/dt') = ɣmv. So, p=ɣmv, which is the given formula for special relativistic momentum (though usually u is used instead of v to distinguish velocities of frames and objects). I get all of this.

Hmm...this doesn't make a lot of sense to me. I think either someone gave you a bad explanation or your notes on it are garbled.

Or, basically, x'=x/ɣ and t'=t/ɣ

But this isn't how the Lorentz transformation actually works.

There are lots of different ways of approaching relativistic energy and momentum. If the approach your instructor used isn't working for you (or for me), why not just look at a different derivation? For example, one can get there by requiring that the results of collisions make sense in all frames of reference: http://www.lightandmatter.com/html_books/6mr/ch01/ch01.html#Section1.3

 

[bobc2]