A New Look at Momentum — Question 3

Question 3

A full velocity reversal requires reversing the relativistic momentum

A particle is turned from $+0.60c$ to $-0.60c$ along x by an external interaction lasting laboratory time $\Delta t$. Use $\mathbf p=\gamma m\mathbf v$, $\gamma=(1-v^2/c^2)^{-1/2}$, and $\mathbf F=d\mathbf p/dt$. Here m is invariant mass.

1. Find the signed impulse.

2. Find the mean laboratory force and compare with a classical estimate using the same endpoint speeds.

Question 3 - Solution

A full velocity reversal requires reversing the relativistic momentum

(a) Method and calculation

$$p_i=1.25m(.60c)=.750mc,\quad p_f=-.750mc.$$$$\boxed{J_x=p_f-p_i=-1.50mc}.$$

(b) Interpretation and further calculation

$$\boxed{\overline F_x=-1.50mc/\Delta t}.$$$$J_{\mathrm{cl}}=m(-.60c-.60c)=-1.20mc,\quad\boxed{J_x/J_{\mathrm{cl}}=1.25}.$$

Physical check

Equal speed magnitudes do not imply zero impulse; momentum is a signed vector.