Question

You have just been pulled over for running a red light, and

the police officer has informed you that the fine will be \(250. In

desperation, you suddenly recall an idea that your physics professor recently discussed in class. In your calmest voice, you tell the officer that the laws of physics prevented you from knowing that the light was red. In fact, as you drove toward it, the light was Doppler shifted to where it appeared green to you. “OK,” says the officer, “Then I’ll ticket you for speeding. The fine is \)1 for every 1 km/h over the posted speed limit of 50 km/h.< How big is your fine? Use 650 nm as the wavelength of red light and 540 nm as the wavelength of green light.

Short answer

The fine will be of$\$200million.$

Step-by-step solution

Given data

Due to the observer is approaching towards the light source, the red light source that has a wavelength of ${\lambda }_o=650nm$experiences a Doppler shift to green light of wavelength $\lambda =540nm.$

Determination of fine amount

The Doppler shift for a light wave approaching, the wavelength

$$\begin{gathered} \lambda =\sqrt{\frac{1-{\displaystyle \frac{v_s}{c}}}{1+\frac{v_s}{c}}}{\lambda }_o \\ 540nm=650nm\sqrt{\frac{c-v_s}{c+v_s}} \\ 0.83=\sqrt{\frac{c-v_s}{c+v_s}} \\ \frac{c-v_s}{c+v_s}=0.{83}^2=0.69 \\ \left(c+v_s\right)0.69=\left(c-v_s\right) \\ 1.69v_s=0.31c \\ {v}_s=0.183\times \left(3\times {10}^8\right)m/s \\ {v}_s=5.5\times {10}^{7}m/s=5.5\times {10}^{4}km/s \\ {v}_s=2\times {10}^{8}km/h \end{gathered}$$

Thus, the speed exceeds $50km/h$by role="math" localid="1649957842651" $\left(2\times {10}^{8}km/h-50km/h\right)$.

Therefore, the fine will berole="math" localid="1649958140230" $\left(2\times {10}^{8}km/h-50km/h\right)\times \$1=\$200million$.