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

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

Step-by-step explanation:

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

`\displaystyle \bar{v} = (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* 10^(3)\;m`.

`\begin{aligned}\displaystyle \text{Percentage Uncertainty in }s &= \frac{\text{Absolute Uncertainty in } s}{\text{Measured Value of }s}* 100\% \\ &=\rm(29\; m)/(42.238* 10^(3)\;m)* 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* 3600 + 31* 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}* 100\% \\ &=\rm(46\; s)/(9106\;s)* 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.