- #1
etotheipi
I'm making a mistake somewhere, I hoped someone could point it out? Say a particle of mass ##2m##, initially at rest, disintegrates into two particles of mass ##m## that move at ##v_{1x} = v## and ##v_{2x} = -v##. Then we switch to a frame moving at ##-U## along the ##x## axis, so that$$v'_{1x} = \frac{U+v}{1+Uv}$$ $$v'_{2x} = \frac{U-v}{1-Uv}$$The total momentum in S' is$$\begin{align*}p_x' = \frac{\frac{U+v}{1+Uv}}{\sqrt{1-\left( \frac{U+v}{1+Uv} \right)^2}}m + \frac{\frac{U-v}{1-Uv}}{\sqrt{1-\left( \frac{U-v}{1-Uv} \right)^2}}m &= \frac{2mU}{\sqrt{(1-v^2) -U^2(1-v^2)}} \\ &= \frac{2mU}{\sqrt{(1-v^2)(1-U^2)}} \\ &= \frac{1}{\sqrt{1-v^2}} [2m \gamma_U U]\end{align*}$$Why does this not equal ##2m \gamma_U U##, but only approaches it in the limit ##v \rightarrow 0##? The initial momentum in S' would surely be ##2m \gamma_U U##, wouldn't it - surely the 3-momentum would be conserved? Please let me know if I've done something dumb. Thanks!