Question

How fast should a person drive his car so that the red signal of light appears green?
(Wavelength for the red color = $6200\mathop {\text{A}}\limits^ \circ$ and Wavelength for green color = $5400\mathop {\text{A}}\limits^ \circ$)
$(A)1.5 \times {10^8}m/s$
$(B)7 \times {10^7}m/s$
$(C)3.9 \times {10^7}m/s$
$(D)2 \times {10^8}m/s$

Answer 1

The principle of the Doppler effect is used here. Doppler effect means the difference between wavelengths or velocities.
The ratio of the velocity of the car to the velocity of light has to be calculated. This ratio equals the ratio of wavelength difference between two colors to the wavelength of red. From this, the velocity of the car is found since we know the velocity of light.

Formula used:
$\dfrac{{{\lambda _{red}} - {\lambda _{green}}}}{{{\lambda _{red}}}} = \dfrac{v}{c}$
$v =$ the velocity of the car,
$c =$ the velocity of the light.

Complete step-by-step solution:
The difference between the two colors i.e. the red and green is
${\lambda _{red}} - {\lambda _{green}} = 6200 - 5400 = 800\mathop {\text{A}}\limits^ \circ$
The ratio of the velocity of the car to the velocity of light is, $\dfrac{v}{c}$
The doppler effect on wavelength gives,
$\dfrac{{{\lambda _{red}} - {\lambda _{green}}}}{{{\lambda _{red}}}} = \dfrac{v}{c}$, $c = 3 \times {10^8}m/s$ (speed of light)
$\Rightarrow v = 3 \times {10^8} \times \dfrac{{800}}{{6200}}$
$\Rightarrow v = \,3.87 \times {10^7}$
$\Rightarrow v \approx 3.9 \times {10^7}$
So, the person should drive the car at a speed of $3.9 \times {10^7}m/s$
Hence the right answer is in option (C).

Note: Generally, we hear about the doppler effect of sound. The Doppler effect or Doppler shift is the change in frequency of a wavelength concerning an observer who is moving to the wave source.
The Doppler effect occurs with almost all types of waves, not just sound. Light waves can be influenced by the speed of the observer in the same way. If one drives fast enough, he can change a red light to occur green to the driver.
Here we take the speed of light to be $c = 3 \times {10^8}m/s$, then the person has to drive $3.9 \times {10^7}m/s$ to shift a red light to look green. On the other hand, it can be said that he needs to travel $18.3\%$ at the speed of light.