Question

A cop pulls you over and asks what speed you were going. “Well, officer, I cannot tell a lie: the speedometer read $\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}\mathbf{}\mathbf{m}\mathbf{/}\mathbf{s}$.” He gives you a ticket, because the speed limit on this highway is $\mathbf{2}\mathbf{.}\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}\mathbf{}\mathbf{m}\mathbf{/}\mathbf{s}$ . 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?

Short answer

The driver is innocent.

Step-by-step solution

Expression for the relationship between proper velocity and ordinary velocity:

Write the relationship between proper and ordinary velocity.

$\mathit{u}\mathbf{=}\frac{\mathbf{1}}{{\sqrt{\mathbf{1}\mathbf{-}{\displaystyle \frac{{\mathbf{\eta }}^{\mathbf{2}}}{{\mathbf{c}}^{\mathbf{2}}}}}}^{\mathbf{\eta }}}$ …… (1)

Here, $\eta$is the proper velocity, u is the ordinary velocity, and c is the speed of light.

Determine that the driver is guilty or innocent:

Substitute $4\times {10}^8\mathrm{m}/\mathrm{s}$for$\eta$ and$3\times {10}^8\mathrm{m}/\mathrm{s}$ for$c$ in equation (1).

$$\begin{gathered} \mathrm{u}-\frac{1}{\sqrt{1-{\displaystyle \frac{{\left(4\times {10}^8\mathrm{m}/\mathrm{s}\right)}^2}{{\left(3\times {10}^8\mathrm{m}/\mathrm{s}\right)}^2}}}}\left(4\times {10}^8\mathrm{m}/\mathrm{s}\right) \\ \mathrm{u}-\left(0.6\right)\times \left(4\times {10}^2\mathrm{m}/\mathrm{s}\right) \\ \mathrm{u}-2.4\times {10}^8\mathrm{m}/\mathrm{s} \end{gathered}$$

As the ordinary velocity is less than the speed light $\left(u<s\right)$, the driver maintained his car’s speed, and hence lawyer proved him innocent.

Therefore, the driver is innocent.