Problem — Human stopwatch

Problem

A classmate times by hand the passage of a car between two markers $10\ \text{m}$ apart. Given that the typical reaction time is $0{,}2\ \text{s}$, estimate the maximum speed beyond which hand timing becomes useless for achieving a relative error $\le 10\%$.

Solution

The dominant uncertainty is the reaction time $\Delta t \approx 0{,}2\ \text{s}$. The relative error on the time (and hence on the speed, since $v = d/t$) is $\Delta t / t$. Imposing the threshold: $\frac{\Delta t}{t} \le 10\% \quad\Rightarrow\quad t \ge \frac{\Delta t}{0{,}10} = \frac{0{,}2}{0{,}10} = 2\ \text{s}$

The transit time must be at least $2\ \text{s}$. With distance $d = 10\ \text{m}$, the corresponding speed is: $v = \frac{d}{t} \le \frac{10}{2} = 5\ \frac{\text{m}}{\text{s}} \approx 18\ \frac{\text{km}}{\text{h}}$

Beyond about $5\ \text{m/s}$ the transit time drops below $2\ \text{s}$ and the relative error exceeds $10\%$: hand timing becomes useless and a photogate is needed.

$\ev{v_\text{max} \approx 5\ \text{m/s} \approx 18\ \text{km/h}}$

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Topics: Experimental method Skills: Error propagation · Fermi estimates