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Chapter 5: Problem 11

What happens to the relativistic Doppler effect when relative velocity is zero? Is this the expected result?

Short Answer

Expert verified
When the relative velocity between the source and the observer is zero (v=0), the relativistic Doppler effect formula simplifies to \(f' = f\), meaning the observed frequency (f') is equal to the emitted frequency (f). This result is expected, as there should be no shift in the frequency due to the Doppler effect when there is no relative motion between the source and the observer.

Step by step solution

01

Understanding the relativistic Doppler effect

The relativistic Doppler effect occurs when there is a relative motion between a source emitting waves and an observer. The frequency observed by the observer (f') is different from the frequency emitted by the source (f), and the difference is due to the relative velocity (v) between them. The relativistic Doppler effect formula is given by: \(f' = f \sqrt{\frac{1 + \frac{v}{c}}{1 - \frac{v}{c}}}\) where c is the speed of light.
02

Set the relative velocity to zero

We are given that the relative velocity between the source and the observer is zero, hence: v = 0 Now, let's substitute v = 0 into the relativistic Doppler effect formula.
03

Calculate the observed frequency

Using the formula, we'll calculate the observed frequency (f'): \(f' = f \sqrt{\frac{1 + \frac{0}{c}}{1 - \frac{0}{c}}}\) After simplifying the expression, we get: \(f' = f \sqrt{\frac{1}{1}}\) Which further simplifies to: \(f' = f\)
04

Interpret the result

The observed frequency (f') in this case is equal to the emitted frequency (f). This means that when there is no relative motion between the source and the observer, the Doppler effect is not present, and the observer perceives the same frequency as the source.
05

Conclude whether the result is expected or not

Based on the analysis and calculations, the result is indeed expected. When there is no relative motion between the source and the observer, it makes sense that there should be no shift in the frequency due to the Doppler effect. In this case, the relativistic Doppler effect is in agreement with our intuition that no frequency change should be observed if there is no relative motion between the source and the observer.

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Most popular questions from this chapter

Show that the relativistic form of Newton's second law is (a) \(F=m \frac{d u}{d t} \frac{1}{\left(1-u^{2} / c^{2}\right)^{3 / 2}} ;\) (b) Find the force needed to accelerate a mass of \(1 \mathrm{kg}\) by \(1 \mathrm{m} / \mathrm{s}^{2}\) when it is traveling at a velocity of \(c / 2\).

(a) Calculate the speed of a 1.00 - \(\mu\) g particle of dust that has the same momentum as a proton moving at \(0.999 c\) (b) What does the small speed tell us about the mass of a proton compared to even a tiny amount of macroscopic matter?

(a) What is \(\gamma\) if \(v=0.100 c ?\) (b) If \(v=0.900 c ?\)

Relativistic effects such as time dilation and length contraction are present for cars and airplanes. Why do these effects seem strange to us?

If two spaceships are heading directly toward each other at \(0.800 c,\) at what speed must a canister be shot from the first ship to approach the other at \(0.999 c\) as seen by the second ship?
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