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Question 12.27: A cop pulls you over and asks what speed you were going. “We...

A cop pulls you over and asks what speed you were going. “Well, officer, I cannot tell a lie: the speedometer read 4 \times 10^{8} m/s.” He gives you a ticket, because the speed limit on this highway is 2.5 \times 10^{8} m/s. In court, your lawyer (who, luckily, has studied physics) points out that a car’s speedometer measures proper velocity, whereas the speed limit is ordinary velocity. Guilty, or innocent?

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Use the result of Problem 12.24(a): u=\frac{1}{\sqrt{1+\eta^{2} / c^{2}}} \eta . \text { Here } \frac{\eta}{c}=\frac{4}{3}, \text { so } \frac{1}{\sqrt{1+16 / 9}}=\frac{3}{5} , and hence u=\frac{3}{5}\left(4 \times 10^{8}\right)=2.4 \times 10^{8} m / s .   Innocent.

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