The Postulates — Question 2

Question 2

Closing speed in a laboratory is not a local light measurement

A detector starts at $x=D$ and moves toward a pulse emitted at the origin at $t=0$, with $x_D=D-vt$. Let $D=900\,\mathrm m$, $v=0.50c$, and $c=3.00\times10^8\,\mathrm{m/s}$.

1. Find the laboratory interception time and position.

2. Find the laboratory rate of decrease of separation and distinguish it from the detector-frame light speed.

Question 2 - Solution

Closing speed in a laboratory is not a local light measurement

(a) Method and calculation

$$ct=D-vt\quad\Rightarrow\quad t=\frac{D}{c+v}.$$$$\boxed{t=2.00\,\mu\mathrm s},\qquad\boxed{x=600\,\mathrm m}.$$

(b) Interpretation and further calculation

$$g(t)=D-(c+v)t,\qquad\boxed{-\dot g=1.50c}.$$$$\text{One local inertial detector frame:}\qquad\boxed{|u\prime_{\mathrm{light}}|=c}.$$

Physical check

The closing rate subtracts two positions at one laboratory time; it is not the speed of one object measured in the rest frame of the other.