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convert-ms-to-kmhr-dimensionally

🚀 Dimensional analysis is a technique used in physics and engineering to convert units from one system to another. In this case, we will focus on converting speed from meters per second (m/s) to kilometers per hour (km/h). This conversion is essential in various applications, such as calculating the speed of vehicles, understanding motion, and analyzing data in scientific experiments. The key to dimensional analysis is to use conversion factors that relate the units involved, allowing us to cancel out units and arrive at the desired measurement.

Theory Explanation

Understanding the Units

To convert from meters per second to kilometers per hour, we need to understand the relationship between these units. 1 kilometer is equal to 1000 meters, and 1 hour is equal to 3600 seconds. Therefore, we can set up our conversion using these relationships.

\[ 1 \text{ km} = 1000 \text{ m}, \quad 1 \text{ h} = 3600 \text{ s}. \]
Setting Up the Conversion Factor

We can create a conversion factor to convert m/s to km/h. The conversion factor will be: \( \frac{1 \text{ km}}{1000 \text{ m}} \) and \( \frac{3600 \text{ s}}{1 \text{ h}} \). By multiplying the speed in m/s by these factors, we can convert the units accordingly.

\[ \text{Speed in km/h} = \text{Speed in m/s} \times \frac{1 \text{ km}}{1000 \text{ m}} \times \frac{3600 \text{ s}}{1 \text{ h}}. \]
Performing the Calculation

After setting up the conversion, we can perform the calculation. We multiply the speed in m/s by the conversion factors to get the speed in km/h. Make sure to cancel out the units properly to ensure the calculation is correct.

Key Points

  • 🎯 1 kilometer = 1000 meters
  • 🎯 1 hour = 3600 seconds
  • 🎯 To convert m/s to km/h, multiply by (3600/1000)

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Examples:💡

Convert 10 m/s to km/h.

Solution:

Step 1: Set up the conversion: 10 m/s × (1 km / 1000 m) × (3600 s / 1 h)

\[ 10 \text{ m/s} \times \frac{1 \text{ km}}{1000 \text{ m}} \times \frac{3600 \text{ s}}{1 \text{ h}}. \]

Step 2: Calculate the conversion: 10 × (3600 / 1000) = 36 km/h

Convert 25 m/s to km/h.

Solution:

Step 1: Set up the conversion: 25 m/s × (1 km / 1000 m) × (3600 s / 1 h)

\[ 25 \text{ m/s} \times \frac{1 \text{ km}}{1000 \text{ m}} \times \frac{3600 \text{ s}}{1 \text{ h}}. \]

Step 2: Calculate the conversion: 25 × (3600 / 1000) = 90 km/h

Common Mistakes

  • Mistake: Forgetting to convert both the numerator and denominator when setting up the conversion factor.

    Correction: Always ensure that you include both the distance and time conversions in your calculation.

  • Mistake: Not canceling out the units properly, leading to incorrect final units.

    Correction: Double-check that the units cancel out correctly to ensure you end up with the desired units.

  • Mistake: Confusing the conversion factors, such as using 60 seconds instead of 3600 seconds for an hour.

    Correction: Remember that 1 hour is equal to 3600 seconds, and use this in your conversion.

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