Relativity - Doppler effect in reflection from a moving mirror

In summary, the question is asking for the frequency of light reflected in a plane mirror moving away from an observer at a uniform speed. The frequency is given by f = f_0 \left(\frac{c-v}{c+v}\right), with c representing the speed of light and v representing the speed of the mirror. The method used to find this frequency does not require special relativity.
  • #1
quantumstate
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Homework Statement



A plane mirror is moving at a uniform speed [tex]v[/tex] away from the observer and a pulsing source of light of frequency [tex]f_0[/tex].

What is the frequency of the light reflected in the mirror as seen by the observer?

This isn't an actual homework question but I have a similar question in which there is ambiguity as to whether the mirror is stationary or not. I get a different answer if the mirror moves I am hoping to comfirm that this is correct but considering this simple case.

Homework Equations



The Attempt at a Solution



This doesn't seem to require any special relativity with my method.

Take [tex]t=0, x=0[/tex] to be the time a pulse is received. At time [tex]t=\frac{1}{f_0}[/tex] the second pulse arrives at x=0. The second pulse catches up with the spaceship in a time given by:

[tex]c\Delta t=v\Delta t+\frac{v}{f_0}[/tex]
[tex]\Delta t = \frac{2v}{f_0(c-v)}[/tex]
[tex]\implies t = \frac{1}{f_0} + \frac{2v}{f_0(c-v)}[/tex]
[tex] t = \frac{1}{f_0} \left(\frac{2v+c-v}{c-v}\right)[/tex]
[tex] \implies f=f_0 \left(\frac{c-v}{c+v}\right)[/tex]
 
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  • #2
So, the frequency of the light reflected in the mirror as seen by the observer is f = f_0 \left(\frac{c-v}{c+v}\right).
 

1. What is the Doppler effect in reflection from a moving mirror?

The Doppler effect in reflection from a moving mirror refers to the change in frequency of a wave when it is reflected off of a mirror that is in motion. This effect is similar to the well-known Doppler effect observed with sound waves, where the perceived frequency of the sound changes depending on the relative motion between the source of the sound and the observer.

2. How does the Doppler effect in reflection from a moving mirror occur?

The Doppler effect in reflection from a moving mirror occurs because of the relative motion between the mirror and the source of the wave. As the mirror moves towards or away from the source, the distance between them changes, causing a change in the perceived frequency of the reflected wave.

3. Can the Doppler effect in reflection from a moving mirror be observed with light waves?

Yes, the Doppler effect in reflection from a moving mirror can be observed with light waves. This was first demonstrated by the French physicist Hippolyte Fizeau in 1851. Today, this effect is commonly used in various applications such as measuring the speed of moving objects and in radar systems.

4. How does the speed of the mirror affect the Doppler effect in reflection?

The speed of the mirror has a direct effect on the amount of frequency shift observed in the reflected wave. The faster the mirror moves, the greater the change in frequency will be. This is because the distance between the mirror and the source of the wave changes more quickly, resulting in a larger change in perceived frequency.

5. Is the Doppler effect in reflection from a moving mirror affected by the angle of incidence?

Yes, the angle of incidence does affect the Doppler effect in reflection from a moving mirror. When the angle of incidence is perpendicular to the mirror's surface, there is no change in frequency observed. However, as the angle of incidence deviates from perpendicular, the amount of frequency shift increases, with the maximum shift occurring at a 45-degree angle of incidence.

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