Applications of Special Relativity

Problem : A train is moving directly towards you at 2×10⁸ m/s. The (monochromatic) light on the front of the train has a wavelength of 250 nanometers in the frame of the train. What wavelength do you observe?

Solution for Problem 1 >>

Using c = fλ we find the frequency of the emitted light to be 1.2×10¹⁵ Hz. The observed frequency is given by:

f = f' = 1.2×1015 = ×1.2×1015 = 2.68×1015

Thus the wavelength is λ = c/f = 3.0×10⁸/2.68×10¹⁵ = 112 nanometers.

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Problem : Light that is assumed to be from the 22.5 cm microwave Hydrogen line is measured at a frequency of 1.2×10³ MHz. How fast is the galaxy from which this light was emitted receding from the earth?

Solution for Problem 2 >>

This is the famous 'redshift' effect. We know that the ratio = . Because f = c/λ this must be equal to the ratio , where the unprimed symbols denoted the frequencies and wavelengths measured on earth. Thus = , where c/(1.2×10⁹) = 25 . Thus:

1.23 = âá’1.23 - 1.23v/c = 1 + v/câá’0.23 = 2.23v/câá’v = 0.105c

This is about 3.15×10⁷ m/s.

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Problem : Consider two ultra-high-speed drag racers. One drag racer has a red stripe on the side and overtakes the other drag racer at a relative speed of c/2 . If the red stripe has wavelength 635 nanometers, what color is the stripe as observed by the other drag racer (that is, what is the wavelength) at the exact instant the overtaking occurs as measured in the frame of the racer-being-overtaken?

Problem : In the previous problem, what is the observed color of the stripe at the instant the overtaken drag racer observes herself being overtaken?

Problem : Explain (qualitatively if you like) why an observer moving in a circle around a stationary source observes the same Doppler effect as one of the transverse cases discussed in Section 1. Which one and what is the frequency shift? Use the fact that if an inertial observer observes the clock of an accelerating object, it is only the instantaneous speed which is important in calculating the time dilation.