Question

*20. Imagine driving down a street toward a traffic light. How fast would you have to go so that the red light (700 nm) would appear green (500 nm)? What fraction of the speed of light c (c = 300,000 km/s) is this velocity?

Answer 1

The observed wavelength, $\lambda$ obs = 500 nm
The wavelength of the source, $\lambda$ source = 700 nm
$\lambda$ obs = $\lambda$ source * SQRT[(1 - _$\beta$ ) / (1 + _$\beta$ )
Where _$\beta$ = v/c, v is the speed of the driver and c is the speed of light
500 = 700 * SQRT[(1 - _$\beta$ ) / (1 + _$\beta$ )
SQRT[(1 - _$\beta$ ) / (1 + _$\beta$ ) = 5/7 = 0.714
Squaring on both the sides,
(1 - _$\beta$ ) / (1 + _$\beta$ ) = 0.51
1 - _$\beta$ = 0.51 + 0.51 * _$\beta$
0.49 = 1.51 * _{$\beta$
$\beta$} = 0.324
v/c = 0.324
v = 0.324 * c
= 0.324 * 300,000 km/s
v = 9.729 * 10⁴ km/s
v/c = 0.324