Question

A physicist drives through a stop light. When he is pulled over, he tells the police officer that the Doppler shift made the red light of wavelength 635 nm appear green to him, with a wavelength of 550 nm. The police officer writes out a traffic citation for speeding. How fast was the physicist traveling, according to his own testimony

Answer 1

Step-by-step explanation:

`\lambda` = Observed wavelength = 550 nm

`\lambda'` = Actual wavelength = 635 nm

c = Speed of light =
`3* 10^8\ \text{m/s}`

v = Velocity of the physicist

Doppler shift is given by

`f=\sqrt{(c+v)/(c-v)}f'\\\Rightarrow (c)/(\lambda)=\sqrt{(c+v)/(c-v)}(c)/(\lambda')\\\Rightarrow (\lambda'^2)/(\lambda^2)=(c+v)/(c-v)\\\Rightarrow (\lambda'^2)/(\lambda^2)=(1+(v)/(c))/(1-(v)/(c))\\\Rightarrow (\lambda'^2)/(\lambda^2)(1-(v)/(c))=1+(v)/(c)\\\Rightarrow (\lambda'^2)/(\lambda^2)(1-(v)/(c))=1+(v)/(c)\\\Rightarrow (\lambda'^2)/(\lambda^2)-1=(v)/(c)(1+(\lambda'^2)/(\lambda^2))\\\Rightarrow v=(c((\lambda'^2)/(\lambda^2)-1))/(1+(\lambda'^2)/(\lambda^2))`

`\Rightarrow v=(3* 10^8* ((635^2)/(550^2)-1))/(1+(635^2)/(550^2))\\\Rightarrow v=42817669.77\ \text{m/s}`

The physicist was traveling at a velocity of
`42817669.77\ \text{m/s}`.