Convert m/s^2 to k/hr^2 using dimensional analysis.

To convert meters per second squared (m/s²) to kilometers per hour squared (km/h²) using dimensional analysis, we'll proceed step by step.

### 1. Conversion from meters to kilometers:
We know that:
\[ 1 \text{ kilometer (km)} = 1000 \text{ meters (m)} \]

Thus, converting meters to kilometers:
\[
1 \text{ m} = 0.001 \text{ km}
\]

### 2. Conversion from seconds squared to hours squared:
We know that:
\[ 1 \text{ hour (h)} = 3600 \text{ seconds (s)} \]

Thus, converting seconds squared to hours squared:
\[
1 \text{ s}^2 = (3600)^2 \text{ h}^2 = 12,960,000 \text{ h}^2
\]

### 3. Combine the conversions:
Now, let's combine both conversions to change units from m/s² to km/h².

\[
1 \frac{\text{m}}{\text{s}^2} = \frac{0.001 \text{ km}}{1 \text{s}^2} = \frac{0.001 \text{ km}}{12,960,000 \text{ h}^2}
\]

Thus:
\[
1 \frac{\text{m}}{\text{s}^2} = 2.7778 \times 10^{-10} \frac{\text{km}}{\text{h}^2}
\]

### Final Result:
\[
1 \, \text{m/s}^2 = 2.7778 \times 10^6 \, \text{km/h}^2
\]

### Explanation:
For every \(1 \, \text{m/s}^2\), it is equivalent to \(12.96 \times 10^6 \, \text{km/h}^2\), accounting for the difference in scaling between meters and kilometers, and seconds to hours squared.

Would you like further details or clarifications?

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Here are five related questions:

1. How would you convert velocity from m/s to km/h?
2. What is the significance of dimensional analysis in unit conversions?
3. Can you explain the difference between linear and squared unit conversions?
4. How would you convert acceleration from km/h² to m/s²?
5. Can this method be used for other unit systems, like miles per hour squared?

**Tip:** Always ensure that both the numerator and denominator are converted appropriately when dealing with squared or cubed units.

Convert m/s^2 to km/h^2 using dimensional analysis.

Learn how to convert acceleration from m/s^2 to km/h^2 using dimensional analysis. This problem involves unit conversion from meters to kilometers and seconds squared to hours squared, resulting in 1 m/s^2 = 2.7778 x 10^6 km/h^2. Suitable for students in Grades 9-12.

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