Rubber Ducky
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Homework Statement
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A particle of mass m is moving in the +x-direction with speed u and has momentum p and energy E in the frame S.
(a) If S' is moving at speed v, find the momentum p' and energy E' in the S' frame.
(b) Note that E' \neq E and p' \neq p, but show that (E')^2-(p')^2c^2=E^2-p^2c^2
Homework Equations
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E=p^2c^2+m^2c^4
E'=\gamma mc^2=\frac{mc^2}{\sqrt{1-\frac{(u')^2}{c^2}}}
p'=\gamma mu'=\frac{mu'}{\sqrt{1-\frac{(u')^2}{c^2}}}
u'=\frac{u-v}{1-\frac{uv}{c^2}}
The Attempt at a Solution
My prof says we need to use (a) to derive (b). It seems like it'd be easier a different way, but whatever. My strategy has been to show that (E')^2-(p')^2c^2=m^2c^4, which would complete the derivation.
I tried playing around with some algebra and got (E')^2=\frac{m^2c^4(c^2-uv)^2}{(c^2-u^2)(c^2-v^2)} and (p')^2=\frac{m^2(u-v)^2}{(1-\frac{uv}{c^2})^2(c^2-u^2)(c^2-v^2)}
I got a little overwhelmed with the algebra at this point. I'd like to know if this is right so far, at least, before trying anything else.