Choose the emitted beam direction as positive

x

. In the original laboratory frame the sensor moves with velocity

- 0.2 c

; let the laser recede from it with velocity

- β c

. The collinear relativistic Doppler shift from source to laboratory and then to sensor gives

\frac{ν _{sensor}}{ν _{0}} = \frac{1.2}{0.8} \frac{1 - β}{1 + β} .

(1)

At the detection threshold this ratio is

633/1000

. Hence

q = \frac{( 633/1000 ) ^{2}}{1.5} = 0.267126, β = \frac{1 - q}{1 + q} ≃ 0.578375.

(2)

The minimum collinear speed is therefore about $0.578 c$ , away from the approaching sensor, while the laser is still aimed towards it. The source must recede fast enough to overcome the sensor's approach-induced blueshift. A larger recession speed gives a longer received wavelength, if the detector has no additional upper cutoff.

As a check, the required relative recession speed is