A car travels a distance 166m +-2m in 5.2s +-0.1. What is the best estimate of the speed of the car

To find the best estimate of the speed of the car, we can use the formula for speed:

\[
\text{Speed} = \frac{\text{Distance}}{\text{Time}}
\]

Given:
- Distance \( d = 166 \, \text{m} \pm 2 \, \text{m} \)
- Time \( t = 5.2 \, \text{s} \pm 0.1 \, \text{s} \)

### Best estimate of speed:
\[
\text{Speed} = \frac{166 \, \text{m}}{5.2 \, \text{s}} \approx 31.92 \, \text{m/s}
\]

Now, we calculate the uncertainty in the speed.

### Uncertainty in speed:

The relative uncertainty of the speed is the sum of the relative uncertainties of the distance and time:

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

Substituting values:
- Uncertainty in distance \( \Delta d = 2 \, \text{m} \)
- Uncertainty in time \( \Delta t = 0.1 \, \text{s} \)

\[
\frac{\Delta v}{v} = \frac{2}{166} + \frac{0.1}{5.2} \approx 0.01205 + 0.01923 = 0.03128
\]

Now, multiply this by the best estimate of speed:

\[
\Delta v = 0.03128 \times 31.92 \, \text{m/s} \approx 0.998 \, \text{m/s}
\]

### Final result:

\[
\text{Speed} = 31.92 \, \text{m/s} \pm 1.00 \, \text{m/s}
\]

Thus, the best estimate of the speed of the car is \( 31.92 \, \text{m/s} \pm 1.00 \, \text{m/s} \).

Would you like more details or have any questions? 

Here are five related questions you might explore:
1. How does the uncertainty affect the accuracy of the result?
2. How can we reduce uncertainty in both distance and time measurements?
3. What would the speed be if the distance were 10 meters shorter?
4. How would the speed change if the time increased by 0.2 seconds?
5. What is the significance of relative uncertainties in real-world measurements?

**Tip:** Always aim to minimize uncertainties by improving the precision of your measurements for better accuracy in calculated results.

A car travels a distance 166m ±2m in 5.2s ±0.1. What is the best estimate of the speed of the car?

Learn how to calculate the speed of a car traveling 166m in 5.2s with uncertainty measurements in both distance and time. This problem involves applying kinematics and uncertainty principles to find the best estimate of speed, suitable for students in Grades 10-12.

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