Question

A marathon runner completes a 42.238 km course in 2 h, 31 min, and 46 s . There is an uncertainty of 29 m in the distance run and an uncertainty of 3 s in the elapsed time.
What is the percent uncertainty in the average speed?

Answer 1

Answer:

The percentage uncertainty in the average speed is 0.10% (2 sig. fig.)

Explanation:

Consider the formula for average speed $\bar{v}$.

$\displaystyle \bar{v} = \frac{s}{t}$,

where

• $s$ is the total distance, and
• $t$ is the time taken.

The percentage uncertainty of a fraction is the sum of percentage uncertainties in

• the numerator, and
• the denominator.

What are the percentage uncertainties in $s$ and $t$ in this question?

The unit of the absolute uncertainty in $s$ is meters. Thus, convert the unit of $s$ to meters:

$s = \rm 42.238\;km = 42.238\times 10^{3}\;m$.

$\begin{aligned}\displaystyle \text{Percentage Uncertainty in }s &= \frac{\text{Absolute Uncertainty in } s}{\text{Measured Value of }s}\times 100\% \\ &=\rm\frac{29\; m}{42.238\times 10^{3}\;m}\times 100\%\\ &= 0.0687\%\end{aligned}$.

The unit of the absolute uncertainty in $t$ is seconds. Convert the unit of $t$ to seconds:

$t = \rm 2\times 3600 + 31\times 60 + 46 = 9106\;s$

Similarly,

$\begin{aligned}\displaystyle \rm \text{Percentage Uncertainty in }t &= \frac{\text{Absolute Uncertainty in }t}{\text{Measured Value of }t}\times 100\% \\ &=\rm\frac{46\; s}{9106\;s}\times 100\%\\ &= 0.0329\%\end{aligned}$.

The average speed $\bar{v}$ here is a fraction of $s$ and $t$. Both $s$ and $t$ come with uncertainty. The percentage uncertainty in $\bar{v}$ will be the sum of percentage uncertainties in $s$ and $t$. That is:

$\text{Percentage Uncertainty in }\bar{v}\\=(\text{Percentage Uncertainty in } s) + (\text{Percentage Uncertainty in } t)\\ = 0.0687\% + 0.0329\%\\ = 0.010\%$.

Generally, keep

• two significant figures for percentage uncertainties that are less than 2%, and
• one for those that are greater than 2%.

The percentage uncertainty in $\bar{v}$ here is less than 2%. Thus, keep two significant figures. However, keep more significant figures than that in calculations to make sure that the final result is accurate.