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AishaGirl
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Homework Statement
A probe is launched with velocity [itex]v=0.8c[/itex]. A beacon emits a light with wavelength [itex]\lambda=500nm[/itex] in its rest frame. Years later the probe is located by NASA using a telescope, When they measure the light they find the wavelength [itex]\lambda=500nm[/itex] in their rest frame. Is this possible? What is the explanation for this observation?
Homework Equations
Dopple shift equation.
The Attempt at a Solution
Essentially the problem states that [itex]v/v\prime = \lambda\prime/\lambda=1[/itex] I can use the doppler shift equation to find the relationship between [itex]\beta[/itex] and [itex]\theta[/itex]
I'm inclined to think that this is not possible and the light should be redshifted but after working through the question there seems to be some angle whereby the shift balances or cancels out and I don't understand why.
Using the doppler shift equation I can say that [itex]\frac{\lambda\prime}{\lambda} = \frac{\sqrt{1-\beta^2}}{1+\beta \cos\theta} \implies \cos\theta=\frac{1}{\beta}\Big( \sqrt{1-b^2}-1\Big)[/itex]
It's clear that the square root term will always be less than or equal to 1 and so the right side will always be negative given that [itex]\cos\theta < 0[/itex] for [itex]\pi/2 < \theta \leq \pi[/itex]
Then using the taylor expansion it becomes [itex]\beta \rightarrow 0 \implies \cos\theta \approx \frac{1}{\beta}\Big( 1-\frac{1}{2}\beta^2-1\Big)=-\frac{\beta}{2}[/itex]
So there is a time when the probe travels nearly perpendicular to pi/2. If I just plug in [itex]\beta=4/5[/itex] I get [itex]\theta=120^\circ[/itex].
All well and good but I still fail to see where the speed of the probe is taken into account... I cannot see a time where the angle of a probe moving at 0.8c will ever balance to create no shift, the Earth is moving at a fraction of the speed of probe. I don't understand...
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