Chapter 7: Problems I

The Doppler Effect

 
Figure 7.12: Computing Doppler shift.

  1. A rocket sends out flashes of light every 2 seconds in its own rest frame, which you receive every 4 seconds. How fast is the rocket going?

1. First solution: This situation is shown in the first drawing in Figure 7.12. In order to find the hyperbolic angle $\alpha$, draw a horizontal line as shown in the enlarged second drawing, resulting in the system of equations 1) \begin{eqnarray} \tanh\alpha &=& \frac{x}{4-x} \\ (4-x)^2-x^2 &=& 2^2 \end{eqnarray} which is easily solved for $x=\frac32$, so that $\vc=\tanh\alpha=\frac35$.

Second solution: Insert $\lambda=4$ and $\lambda'=2$ into (7) of §6.4, and solve for $\vc$.

1) This method can be be used to derive the Doppler shift formula in general, yielding \begin{equation} \vc = \frac{\lambda^2-\lambda'^2}{\lambda^2+\lambda'^2} \end{equation} which is equivalent to (7) of §6.4; in this example, $\lambda=4$ and $\lambda'=2$.