an athlete ran a distance of 100.0 +- 0.2m in a time of 9.8 +- 0.1s. What was their speed?

To find the speed of the athlete, we need to calculate the speed and its associated uncertainty using the given distance and time measurements.

### Formula for Speed

The speed \( v \) of an object is given by the formula:

\[
v = \frac{d}{t}
\]

where \( d \) is the distance and \( t \) is the time.

### Given Values

- Distance \( d = 100.0 \pm 0.2 \, \text{m} \)
- Time \( t = 9.8 \pm 0.1 \, \text{s} \)

### Calculating Speed

Using the formula:

\[
v = \frac{100.0 \, \text{m}}{9.8 \, \text{s}} \approx 10.2041 \, \text{m/s}
\]

### Calculating Uncertainty in Speed

To calculate the uncertainty in the speed, we use the method of propagation of uncertainties. The formula for the propagation of uncertainties when dividing two quantities is:

\[
\frac{\Delta v}{v} = \sqrt{\left( \frac{\Delta d}{d} \right)^2 + \left( \frac{\Delta t}{t} \right)^2}
\]

Where:

- \(\Delta v\) is the uncertainty in speed
- \(\Delta d = 0.2 \, \text{m}\) is the uncertainty in distance
- \(\Delta t = 0.1 \, \text{s}\) is the uncertainty in time

Substituting the values:

\[
\frac{\Delta v}{10.2041} = \sqrt{\left( \frac{0.2}{100.0} \right)^2 + \left( \frac{0.1}{9.8} \right)^2}
\]

Calculating each term:

- \(\left( \frac{0.2}{100.0} \right)^2 = 0.000004\)
- \(\left( \frac{0.1}{9.8} \right)^2 \approx 0.000104\)

\[
\frac{\Delta v}{10.2041} = \sqrt{0.000004 + 0.000104} = \sqrt{0.000108} \approx 0.01039
\]

Therefore, the relative uncertainty is approximately \(0.01039\).

Calculating \(\Delta v\):

\[
\Delta v = 10.2041 \times 0.01039 \approx 0.1060 \, \text{m/s}
\]

### Final Result

The speed of the athlete, including the uncertainty, is:

\[
v = 10.2 \pm 0.1 \, \text{m/s}
\]

This result indicates that the athlete's speed is \(10.2 \, \text{m/s}\) with an uncertainty of \(0.1 \, \text{m/s}\).

---

Would you like more details or have any questions? Here are some related questions that might interest you:

1. What is the significance of propagation of uncertainties?
2. How do you calculate uncertainties for multiplication or addition operations?
3. Can you explain more about significant figures and their importance in calculations?
4. What are other real-life applications of uncertainty calculations?
5. How do different types of measurement errors affect uncertainty calculations?
6. How would rounding errors impact the final result?
7. Can you provide more examples of using uncertainty in sports?
8. How can uncertainty be reduced in measurements?

### Tip:
Always keep track of your significant figures to ensure your calculated uncertainties accurately reflect the precision of your measurements.

Learn how to calculate the speed of an athlete given a distance of 100.0 meters +- 0.2 meters and a time of 9.8 seconds +- 0.1 seconds, including uncertainty calculations. This problem covers physics concepts such as speed formula and propagation of uncertainty. Suitable for advanced high school students.

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