The Relativity of Length — Question 6

Question 6

Passing a station uses one station clock and two train clocks

A train has proper length $L_0=300\,\mathrm m$ and speed $v=0.80c$ relative to a station. Consider the front and then rear passing the same station point. Use $\gamma=(1-v^2/c^2)^{-1/2}$ and $c=3.00\times10^8\,\mathrm{m/s}$.

1. Find the station-clock interval between the crossings.

2. Find the train-frame coordinate interval and explain which interval is proper for this event pair.

Question 6 - Solution

Passing a station uses one station clock and two train clocks

(a) Method and calculation

$$L=L_0/\gamma=180\,\mathrm m,\quad\boxed{\Delta t=L/v=0.750\,\mu\mathrm s}.$$$$\Delta x=0\quad\Rightarrow\quad\Delta t\prime=\gamma\Delta t.$$

(b) Interpretation and further calculation

$$\boxed{\Delta t\prime=1.250\,\mu\mathrm s=L_0/v}.$$$$\boxed{\Delta\tau=\Delta t=0.750\,\mu\mathrm s}.$$

Physical check

The crossing events occur at one station location, but at opposite train endpoints. The train-frame interval needs synchronized clocks at different positions.