Dimensional Analysis

Introduction

• Question: How far away is the beach from where you live?
  • Americans typically answer in miles.
  • People from other countries usually answer in kilometers.
• Example:
  • Susie (USA): 2 miles
  • Kiara (Australia): 3.2 kilometers
  • Both distances are equivalent but in different units.

Importance of Converting Units

• Different regions use different units.
• Conversion is critical for accurate communication.
• Science:
  • Uses a standardized system of units known as SI units.
  • Kilometers are standard for distance.
  • Miles are a customary unit in the USA.

Conversion Factors

• Definition: A conversion factor allows us to convert from one unit to another.
• Finding Conversion Factors: Can be found online or in textbooks.

Example Conversion

• Problem: Convert 241 kilometers to miles.
• Conversion Factor: 1 km = 0.6214 miles
• Calculation:
  • 241 kilometers * (0.6214 miles / 1 kilometer) = 150 miles
  • Kilometers cancel out, leaving the distance in miles.
• General Formula:
  • Initial units * (Desired units / Initial units) = Desired units

Challenge Problem

• Problem: Convert 60 miles/hour to kilometers/second.
• Conversion Factors:
  • 1 km = 0.6214 miles
  • 1 hour = 60 minutes
  • 1 minute = 60 seconds
• Dimensional Analysis Steps:
  • Start with 60 miles/hour.
  • Convert to kilometers per mile, then kilometers per minute, then kilometers per second.
  • Multiply across the top and divide by the numbers in the bottom row.
• Final Answer: 0.027 kilometers/second

Conclusion

• Converting units can be complex but is essential.
• Practice makes it easier.
• Fun Note: Solving dimensional analysis problems can help you understand how far you need to travel to get to the beach in any unit!