To determine the amount of uncertainty in the timing measured, follow these steps:
1. Identify the given values:
- The recorded time of the 100-meter sprint is 11.5 seconds.
- The uncertainty percentage of the stopwatch is 1.7%.
2. Convert the percentage uncertainty into its decimal form:
- [tex]\( 1.7\% \)[/tex] can be expressed as [tex]\( \frac{1.7}{100} = 0.017 \)[/tex].
3. Calculate the absolute uncertainty in the time:
- Multiply the recorded time by the decimal form of the uncertainty.
[tex]\[
\text{uncertainty amount} = 11.5 \times 0.017 = 0.1955 \text{ seconds}
\][/tex]
4. Determine which of the provided options best matches the calculated uncertainty:
- We'll round the calculated uncertainty amount to the appropriate number of decimal places to match each given option:
- To three decimal places: [tex]\( 0.1955 \)[/tex] rounds to [tex]\( 0.196 \)[/tex].
- To two decimal places: [tex]\( 0.1955 \)[/tex] rounds to [tex]\( 0.20 \)[/tex].
- To four decimal places: [tex]\( 0.1955 \)[/tex] remains [tex]\( 0.1955 \)[/tex].
- To one decimal place: [tex]\( 0.1955 \)[/tex] rounds to [tex]\( 0.2 \)[/tex].
5. Match the calculated uncertainties with the given options:
- Option 1: [tex]\( 11.5 \, s \pm 0.196 \, s \)[/tex] (rounded to three decimal places)
- Option 2: [tex]\( 11.5 \, s \pm 0.19 \, s \)[/tex] (rounded to two decimal places, but actually [tex]\( 0.1955 \)[/tex] to [tex]\( 0.19 \)[/tex])
- Option 3: [tex]\( 11.5 \, s \pm 0.1955 \, s \)[/tex] (rounded to four decimal places)
- Option 4: [tex]\( 11.5 \, s \pm 0.2 \, s \)[/tex] (rounded to one decimal place)
Given these calculations, the uncertainty in the measured time is [tex]\( 0.1955 \, s \)[/tex]. Therefore, the most accurate representation of the uncertainty is:
[tex]\[ 11.5 \, s \pm 0.1955 \, s \][/tex]
Thus, the correct answer is:
[tex]\[ 11.5 s \pm 0.1955 s \][/tex]
