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

The speed is
`v = 8.1 *10^(7) \ m/s`

Step-by-step 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{(1 + (v)/(c) )/(1 - (v)/(c) ) } * \lambda_o`

Here c is the speed of the light

and v is the speed of the physicist

So

`(\lambda_r)/(\lambda_o) = \sqrt{(1 + (v)/(c) )/(1 - (v)/(c) ) }`

=>
`(680nm)/(515nm) = \sqrt{(1 + (v)/(c) )/(1 - (v)/(c) ) }`

=>
`[(680nm)/(515nm) ]^2= (1 + (v)/(c) )/(1 - (v)/(c) )`

=>
`1.743 = (1 + (v)/(c) )/(1 - (v)/(c) )`

=>
`1.743 - 1.743(v)/(c) = 1 + (v)/(c)`

=>
`0.743 = 2.743(v)/(c)`

=>
`v = (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`