A New Look at Momentum — Question 10

Question 10

A large momentum uncertainty can correspond to a small speed uncertainty

A momentum measurement gives $p=(10.0\pm0.1)mc$, where the quoted uncertainty is one standard deviation and m is exactly known. Set $w=p/(mc)$. 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. Derive $d(v/c)/dw$ and evaluate the central speed.

2. Find the first-order speed uncertainty and compare the fractional speed and momentum uncertainties.

Question 10 - Solution

A large momentum uncertainty can correspond to a small speed uncertainty

(a) Method and calculation

$$\beta=\frac w{\sqrt{1+w^2}},\qquad\boxed{d\beta/dw=(1+w^2)^{-3/2}}.$$$$\boxed{\beta=10/\sqrt{101}\simeq0.995037}.$$

(b) Interpretation and further calculation

$$\boxed{\sigma_\beta\simeq\frac{0.1}{101^{3/2}}=9.85185\times10^{-5}}.$$$$\boxed{\sigma_v/v\simeq\frac{0.01}{101}=9.90099\times10^{-5}},\qquad\sigma_p/p=0.0100.$$

Physical check

Near c, increasing momentum changes speed only weakly. The result assumes a narrow distribution for linear uncertainty propagation.