The Relativity of Length — Question 3

Question 3

Length and transit measurements jointly recover proper length

A rod moving longitudinally has laboratory length $L=72.0\,\mathrm m$. It takes $0.400\,\mu\mathrm s$ to pass a fixed point, from front crossing to rear crossing. Use $\gamma=(1-v^2/c^2)^{-1/2}$ and $c=3.00\times10^8\,\mathrm{m/s}$.

1. Infer its speed and Lorentz factor.

2. Find its proper length and distinguish the passing interval from a simultaneous spatial measurement.

Question 3 - Solution

Length and transit measurements jointly recover proper length

(a) Method and calculation

$$v=L/\Delta t=72/(0.400\times10^{-6})=\boxed{0.600c}.$$$$\gamma=1/\sqrt{1-0.600^2}=\boxed{1.250}.$$

(b) Interpretation and further calculation

$$\boxed{L_0=\gamma L=90.0\,\mathrm m}.$$$$\text{Passing events: }\Delta x=0;\qquad\text{length events: }\Delta t=0.$$

Physical check

The two measurements use different pairs of events and provide independent information about speed and rest length.