I am struggling with how to solve these physics problems about speed and the EM spectrum, how do I solve these problems?

Question

1a.You're caught running a red light on Hwy 317. Attempting to impress the skeptical patrol officer with your physics knowledge, you claim that you were traveling so fast that the red light (725 nm) appeared yellow (589 nm) to you. How fast would you have been traveling (in miles per hour) if that had been the case?

1b.If you had been traveling that fast and the light actually was yellow, what color would it have seemed to you?

 -Orange

 -Blue

 -Yellow

 -Invisible (Ultraviolet or Infrared)

 -Red

2a.A galaxy known to be traveling away from the Earth at 0.619 percent of the speed of light is observed on Earth to have a strong peak in its spectrum at 680.9 nm. However, one small portion of the galaxy (a "dwarf") has a similar observed peak with a wavelength that is 4.20 nm shorter. (This situation is not as strange as it may sound. Several galaxies can be observed to be in the process of collision or merger, such as the Whirlpool Galaxy. Several others, particularly radio galaxies, often exhibit high-speed jets or lobes.)

Calculate the wavelength of the light emitted by the bulk of the galaxy.

2b. What is the speed of the dwarf, relative to its "parent" galaxy? (Enter your answer as negative if the dwarf is moving toward Earth relative to the rest of the galaxy.)

Answer 1

1a. Doppler shift v/c ≈ z = Δλ/λ_o = (589 - 725)/725 = -0.187₆ [I keep 1 more figure than number of sig figs to minimize roundoff error]. v ≈ cz = -5.62₄ x 10⁷ m/s (1 mph /0.4470 m/s) = -1.26 x 10⁸ mph towards the light.

[Probably you're not doing the relativistic correction, but just in case: z = [(c+v)/(c-v)]^1/2 - 1 and after algebra, v/c = [1 - (1+z)²] / [1 + (1+z)²] = (1 - .6600) / (1 + .6600) = 0.204₈ and v = (-) 1.37 x 10⁸ mph.]

1b. z = -0.187₆ = (x - 589)/589, x - 589 = -110, x = 479 nm = Blue.

2a. z = 0.00619 = (λ - λ_o) / λ_o, 0.00619 λ_o = 680.9 - λ_o, (1.00619) λ_o = 680.9, λ_o = 676.7 nm.

2b. Relative v/c = (Δλ = -4.20)/680.9 = -0.00616₈, v = -1.84₉ x 10⁶ m/s = -4.14 x 10⁶ mph.