Measuring an Event — Question 5

Question 5

A transponder delay biases naive radar range

A stationary transponder waits $\delta=2.0\,\mathrm{ns}$ after receiving a pulse before sending the reply. The source emits at zero and receives the reply at $22.0\,\mathrm{ns}$. The delay is specified in the common rest frame. Use $c=3.00\times10^8\,\mathrm{m/s}$ and Einstein-synchronized laboratory clocks.

1. Find the actual range and the transponder reception time.

2. Find the error in range and reception time if the delay is ignored.

Question 5 - Solution

A transponder delay biases naive radar range

(a) Method and calculation

$$t_R=2L/c+\delta,\qquad\boxed{L=c(t_R-\delta)/2=3.00\,\mathrm m}.$$$$\boxed{t_{\mathrm{arrive}}=L/c=10.0\,\mathrm{ns}}.$$

(b) Interpretation and further calculation

$$L_{\mathrm{naive}}=ct_R/2=3.30\,\mathrm m,\quad\boxed{\Delta L=0.30\,\mathrm m}.$$$$t_{\mathrm{midpoint}}=11.0\,\mathrm{ns},\quad\boxed{\Delta t=1.0\,\mathrm{ns}}.$$

Physical check

The midpoint lies halfway through the waiting interval; it is neither the arrival nor the departure event.