A radio station broadcasting on a frequency of 106 MHz is received by an astronaut headed toward the station at a speed 4.0×107 m/s. To what frequency must the astronaut's radio be tuned?( use MHz for final unit answers)
Use the relavistic doppler shift formula.
It does not depend upon which object is moving; only the relative velocity matters. The formula can be found at
http://en.wikipedia.org/wiki/Relativistic_Doppler_effect
and probably also in your textbook.
In your case, the relative velocity is 0.133 of the speed of light.
The received frequency will be higher by a factor
sqrt(1.133/0.867) = 1.143
To find the frequency at which the astronaut's radio must be tuned, we can use the concept of the Doppler effect in physics. The Doppler effect describes the change in frequency of a wave (in this case, the radio wave) due to the relative motion between the source (the radio station) and the observer (the astronaut in this case).
The general formula for the Doppler effect with respect to frequency is:
f' = f * (V + Vr) / (V + Vs),
where:
- f' is the observed frequency (the frequency received by the astronaut),
- f is the original frequency (the frequency transmitted by the radio station),
- V is the speed of sound (since radio waves are a form of electromagnetic radiation, and travel at the speed of light, we can assume the speed of sound is equal to the speed of light, c),
- Vr is the velocity of the receiver (in this case, the astronaut),
- Vs is the velocity of the source (in this case, the radio station).
In this scenario, the radio station is at rest, so Vs = 0. The speed of light, c, is approximately 3.0 x 10^8 m/s.
Using the given values, we can substitute them into the formula and solve for f':
f' = f * (c + Vr) / (c + 0),
f' = f * (c + Vr) / c.
Plugging in the known values for c and Vr:
f' = f * (3.0 x 10^8 m/s + 4.0 x 10^7 m/s) / (3.0 x 10^8 m/s).
Now, we convert the obtained frequency to MHz by dividing it by 10^6:
f' = (f * (3.0 x 10^8 + 4.0 x 10^7)) / (3.0 x 10^8) = (f * (30 + 4)) / 30 = (f * 34) / 30.
Therefore, the frequency at which the astronaut's radio must be tuned is approximately:
f' = (34/30) * f = (34/30) * 106 MHz ≈ 118.2 MHz.
So, the astronaut's radio must be tuned to a frequency of approximately 118.2 MHz.