Does going too fast justify things?

Classical Mechanics Level 3

A driver is caught going through a red light. The driver claims to the judge that the color she actually saw was green $\left(\nu = 5.60 \times 10^{14} \text{ Hz} \right)$ and not red $\left(\nu _{0} = 4.80 \times 10^{14} \text{ Hz}\right)$ because of the doppler effect. The judge accepts this explanation and instead fines her for speeding at the rate of $ $1$ for each $\text{ km/h}$ she exceeded the speed limit of $80 \text{ km/h.}$ What is the fine?

$165,000 $165,000,000 $330,000 $330,000,000

1 solution

Brilliant Physics Staff
Nov 2, 2015

A first approach to this problem employs the usual Doppler shift, i.e. $f^\prime = f\left(1+v/c\right)$ .

However, this calculation implies that $v$ is on the order of $c$ , and $c$ is the speed of light, thus our system is fully relativistic.

Near the speed of light we need to apply relativistic mechanics to arrive at the correct expression for Doppler shifts.

This yields $f^\prime = f \sqrt{\frac{1+v/c}{1-v/c}}$ Solving for $v$ using the relativistic form yields $v\approx 4.5 \times 10^7$ m/s or $1.6\times 10^8$ km/hr.

Does going too fast justify things?

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