A distant star is traveling directly toward Earth with a speed of 30,100 km/s.

(a) When the wavelengths in this star's spectrum are measured on Earth, are they greater or less than the wavelengths we would find if the star were at rest relative to us and why?

(b) By what fraction are the wavelengths in this star's spectrum shifted?

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a) wavelengths seen decrease, b) λ'/λ = 0.9998

Explanation:

a) This is a relativistic doppler effect problem that is described by

f' = f₀ √ (1 + v / c) / √ (1 - v / c)

The speed of light be related to wavelength and frequency

c = λ f

f = c / λ

We replace

c / λ' = c / λ √ (1 + v / c) / √ (1 - v / c)

λ' = λ √ [ (1-v / c) / (1 + v / c) ]

This expression gives us the wavelength depends on the speed of the premiere with respect to the Earth, as the estuary approaches the Earth the speed is positive

We can see that at the root it is less than 1 whereby the wavelengths seen decrease

b) we reduce the speed to m / s

v = 30 km / s = 3 10⁴ m / s

Change is

λ'/λ = √ [ (1-v / c) / (1 + v / c) ]

λ' / λ = √ [ (1 - 3 10⁴/3 10⁸) / (1 + 3 10⁴/3 10⁸) ]

λ' / λ = √ (0.9999 / 1.0001)

λ'/λ = 0.9998