Doppler Effect for Light — Question 2

Question 2

Opposite emission directions separate source frequency from speed

A moving monochromatic source is observed along its forward and backward laboratory directions. The frequencies are $f_+=900\,\mathrm{MHz}$ and $f_-=100\,\mathrm{MHz}$.

1. Use $f_\pm=\gamma(1\pm\beta)f_0$ to infer $f_0$.

2. Find the speed and Lorentz factor from the frequency sum and difference.

Question 2 - Solution

Opposite emission directions separate source frequency from speed

(a) Method and calculation

$$f_+f_-=\gamma^2(1-\beta^2)f_0^2=f_0^2.$$$$\boxed{f_0=\sqrt{f_+f_-}=300\,\mathrm{MHz}}.$$

(b) Interpretation and further calculation

$$\boxed{\beta=\frac{f_+-f_-}{f_++f_-}=0.800}.$$$$\boxed{\gamma=\frac{f_++f_-}{2f_0}=5/3}.$$

Physical check

The geometric mean removes the motion while the normalized difference isolates its direction and speed.