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 680 nm appear green to him, with a wavelength of 515 nm. The police officer writes out a traffic citation for speeding. How fast was the physicist traveling, according to his own testimony?

Answer 1

Answer:

The speed is $v = 8.1 *10^{7} \ m/s$

Explanation:

From the question we are told that

   The real  wavelength is  $\lambda_r = 680 \ nm = 680 *10^{-9} \ m$

   The observed  wavelength is $\lambda_o = 515 \ nm = 515 *10^{-9} \ m$

Generally the real wavelength of the stop light according to doppler effect is mathematically as

      $\lambda _r = \sqrt{\frac{1 + \frac{v}{c} }{1 - \frac{v}{c} } } * \lambda_o$

Here c is the speed of the light  

and  v is the speed of the physicist

So

        $\frac{\lambda_r}{\lambda_o} = \sqrt{\frac{1 + \frac{v}{c} }{1 - \frac{v}{c} } }$

=>    $\frac{680nm}{515nm} = \sqrt{\frac{1 + \frac{v}{c} }{1 - \frac{v}{c} } }$

=>    $[\frac{680nm}{515nm} ]^2= \frac{1 + \frac{v}{c} }{1 - \frac{v}{c} }$

=>   $1.743 = \frac{1 + \frac{v}{c} }{1 - \frac{v}{c} }$

=>   $1.743 - 1.743\frac{v}{c} = 1 + \frac{v}{c}$

=>   $0.743 = 2.743\frac{v}{c}$

=>   $v = \frac{0.743c}{2.743}$

=>   $v = 0.2709c$

substituting $3.0*0^{8} \ m/s$  for c

    =>    $v = 0.2709 * 3.0*10^{8}$

    =>    $v = 8.1 *10^{7} \ m/s$