b. If you are cruising down the road at 50 mph and a policeman with a radar gun (wavelength 1 cm) reflects radar light off your bumper, what wavelength of radar light does he get back? Assume the policeman is stationary, and you are moving towards him. Light that gets reflected has twice as much shift as light that is just emitted, so use the formula for the shift of emitted light but multiply by two
To calculate the wavelength of radar light reflected off your bumper, we can use the formula for the Doppler shift. The Doppler shift is the change in frequency (or wavelength) of a wave due to the motion of the source or the observer.
In this scenario, the observer (policeman with the radar gun) is stationary, and you are moving towards him. The formula for the Doppler shift in this case is:
Δλ = (2v/c) * λ
Where:
- Δλ is the change in wavelength
- v is the velocity of the moving object towards the observer
- c is the speed of light in a vacuum
- λ is the wavelength of the emitted light
Since the given information indicates that the radar gun emits light with a wavelength of 1 cm (0.01 m) and you are moving at a velocity of 50 mph (22.35 m/s) towards the policeman, we can plug these values into the formula:
Δλ = (2 * 22.35 / 299,792,458) * 0.01
Calculating this gives us:
Δλ ≈ 0.000000314 m
Therefore, the wavelength of radar light reflected off your bumper would be approximately 0.01 - 0.000000314 m (the emitted wavelength minus the shift):
λ (reflected) ≈ 0.009999686 m or 0.9999686 cm.
So, the policeman would get back radar light with a wavelength of approximately 0.9999686 cm.