Question

Time intervals measured with a stopwatch typically have an uncertainty of about 0.2 s due to human reaction time at the start and stop moments. What is the percent uncertainty of a hand-timed measurement of (a) 5.5 s, (b) 55 s, (c) 5.5 min?

Short answer

1. The percent uncertainty of the hand-timed measurements of 5.5 s is 4%.
2. The percent uncertainty of the hand-timed measurements of 55 s is 0.4%.
3. The percent uncertainty of the hand-timed measurements of 5.5 min is 0.06%.

Step-by-step solution

Writing the formula of percent uncertainty

The uncertainty in measurement is defined as the variation of the values that are assigned to a particular quantity.

The formula of percent uncertainty is

$∆T=\frac{T_2}{T_1}\times 100$ ...... (I)

Calculation of the percent uncertainty for 5.5 s

The uncertainty in measurements with a stopwatch $T_1=0.2\mathrm{s}$.

The given hand-timed measurement is $T_2=5.5\mathrm{s}$.

From equation (i), the percent uncertainty is:

$$\begin{gathered} ∆T=\frac{0.2}{5.5}\times 100 \\ ∆T=3.64\% \\ ∆T\approx 4\% \end{gathered}$$

Thus, the percent uncertainty is 4%.

Calculation of the percent uncertainty for 55 s

The given hand-timed measurement is $T_2=55\mathrm{s}$.

From equation (i), the percent uncertainty is

$$\begin{gathered} ∆T=\frac{0.2}{55}\times 100\% \\ ∆T=0.364\% \\ ∆T\approx 0.4\% \end{gathered}$$

Thus, the percent uncertainty is 0.4%.

Calculation of the percent uncertainty for 5.5 min

The given hand-timed measurement is:

$$\begin{gathered} T_2=5.5\min \\ {\mathrm{T}}_2=5.5\times 60\mathrm{s} \\ {\mathrm{T}}_2=330\mathrm{s} \end{gathered}$$

From equation (i), the percent uncertainty is:

$$\begin{gathered} ∆T=\frac{0.2}{330}\times 100\% \\ ∆T=0.06\% \end{gathered}$$

Thus, the percent uncertainty is 0.06%.