# Applications of Special Relativity

## Contents

#### Problems on the Relativistic Doppler Effect

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?

Using c = 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.

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?

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