Question

Slick Willy is in traffic court (again) contesting a $50.00 ticket for running a red light. “You see, your Honor, as I was approaching the light, it appeared yellow to me because of the Doppler effect. The red light from the traffic signal was shifted up in frequency because I was traveling towards it, just like the pitch of an approaching car rises as it approaches you.”
Calculate how fast Slick Willy must have been driving, in meters per second, to observe the red light (wavelength of 683 nm) as yellow (wavelength of 576 nm). (Treat the traffic light as stationary, and assume the Doppler shift formula for sound works for light as well.)

Answer 1

Answer and Explanation:

$Greetings!$

$I'm~Isabelle~Williams~and~I~will~be~answering~your~question!$

$Given:$

$Wave~length~ of~ red ~ light = 683~ nm$

$Wave ~length ~of~ yellow~ light = 576~ nm$

$The~ speed ~of~ the~ Slick~ Willy~ can~ be:$

$\boxed{~f'=(\frac{v+v_{o} }{v} )f~}$

$Where:$

$f'=observed~frequency$

$f=actual~frequency$

$v=speed~of~light$

$v_{o}=~speed~of~observer$

$Plug~in~the~values:$

$683\times 10^-^9=576\times 10^-^9 ~(1+\frac{v_{o} }{3\times 10^8} )$

$\frac{683}{576}=1+\frac{v_{o} }{3\times 10^8}$

$\frac{v_{o} }{3\times 10^8} =\frac{683}{576}=-1$

$v_{o}=0.186\times 3\times 10^8$

$v_{o}=0.558\times 10^8 ~m/s$

$Hope~this~helps!~Have~an~amazing~day~ahead!~$

$-Isabelle~Williams$