By what factor are the wavelengths in this star's spectrum changed?

Question

A distant star is traveling directly away from Earth with a speed of 40000 km/s  .

By what factor are the wavelengths in this star's spectrum changed?

Answer 1

v = 4 x 10⁷ m/s

At this high speed, you can use the approximation

Δλ/λ₀ = v/c ⇒ Δλ = (v/c)/λ₀

λ = λ₀ + Δλ = λ₀ + (v/c)/λ₀ = λ₀(1 + v/c)

λ/λ₀ = 1 + v/c

Note the factor must be greater than 1, as this situation gives a red shift. Now plug in

v = 4 x 10⁷ m/s

c = 3 x 10⁸ m/s

and get the factor λ/λ₀.