A measurement gives 76 m/s. Decide which measurement is the most accurate and which one is the most precise. A: between 76.8 m/s and 77.2 m/s B: 74 m/s C: 75.5 m/s +or- 0.70% D: 75 m/s +or- 0.1 m/s

To determine which measurement is the most accurate and which one is the most precise, we need to understand the definitions of accuracy and precision:

- **Accuracy** refers to how close a measured value is to the true value.
- **Precision** refers to how close repeated measurements are to each other or how finely a measurement is specified.

Given the measurements and their descriptions:

1. **A: Between 76.8 m/s and 77.2 m/s**  
   This measurement is provided as a range. The midpoint is \( \frac{76.8 + 77.2}{2} = 77.0 \, \text{m/s} \), and the range (precision) is \( \pm 0.2 \, \text{m/s} \).

2. **B: 74 m/s**  
   This is a single value without any indication of uncertainty or range.

3. **C: 75.5 m/s ± 0.70%**  
   The uncertainty in this measurement is \( 0.70\% \) of 75.5 m/s, which equals \( 75.5 \times 0.007 = 0.5285 \, \text{m/s} \). So, the measurement can be expressed as \( 75.5 \, \text{m/s} \pm 0.53 \, \text{m/s} \).

4. **D: 75 m/s ± 0.1 m/s**  
   This measurement has a clear uncertainty, indicating it could be anywhere between \( 74.9 \, \text{m/s} \) and \( 75.1 \, \text{m/s} \).

### Evaluating Accuracy:
- The most accurate measurement is the one closest to the given measurement of 76 m/s.
  - **A**: The midpoint is 77.0 m/s, which is \( |77.0 - 76| = 1.0 \) m/s away from the true value.
  - **B**: 74 m/s is \( |74 - 76| = 2.0 \) m/s away from the true value.
  - **C**: 75.5 m/s is \( |75.5 - 76| = 0.5 \) m/s away from the true value.
  - **D**: 75 m/s is \( |75 - 76| = 1.0 \) m/s away from the true value.

  Therefore, **C** (75.5 m/s ± 0.53 m/s) is the most accurate measurement.

### Evaluating Precision:
- The most precise measurement has the smallest uncertainty or range.
  - **A**: The range is 0.4 m/s (± 0.2 m/s).
  - **B**: No precision is provided (no uncertainty).
  - **C**: The uncertainty is 0.53 m/s (± 0.53 m/s).
  - **D**: The uncertainty is 0.1 m/s (± 0.1 m/s).

  Therefore, **D** (75 m/s ± 0.1 m/s) is the most precise measurement.

### Summary:
- **Most Accurate:** Measurement **C** (75.5 m/s ± 0.53 m/s)
- **Most Precise:** Measurement **D** (75 m/s ± 0.1 m/s)

Would you like more details or have any questions?

Here are five questions that expand on this topic:
1. How does the percentage uncertainty affect the accuracy of a measurement?
2. Why is precision important in scientific measurements?
3. Can a measurement be precise but not accurate? Explain with an example.
4. How can one reduce the uncertainty in a measurement?
5. What methods are used to determine the true value in a measurement?

**Tip:** Always report both the measurement and its uncertainty to provide a complete picture of your data's reliability.

Explore the concepts of measurement accuracy and precision with detailed analysis and comparisons. Learn how to determine the most accurate and precise measurements using examples and calculations. Suitable for high school students and anyone interested in scientific measurements.

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