17. Tommy Wants To Convert 68 Miles Per Hour To Inches Per Second. This Is Hisconversion, Is It Set Up

17. Tommy Wants To Convert 68 Miles Per Hour To Inches Per Second. This Is His Conversion, Is It Set Up

Introduction

Understanding how to convert units of measurement is fundamental in various fields such as engineering, physics, and everyday problem-solving. When Tommy attempts to convert 68 miles per hour (mph) to inches per second (in/sec), he must ensure his conversion setup is accurate. This process involves recognizing the relationships between miles, inches, hours, and seconds, and then applying the correct conversion factors systematically. In this article, we will analyze Tommy's conversion setup step-by-step, verify whether it is correct, and explore the principles behind such unit conversions.

Understanding the Units Involved

Before delving into the conversion process, it’s essential to understand the units involved:

• Miles per Hour (mph): A measure of speed indicating the number of miles traveled in one hour.
• Inches per Second (in/sec): A measure of speed indicating the number of inches traveled per second.

The goal is to convert from miles/hour to inches/second, which involves converting miles to inches and hours to seconds.

Basic Conversion Principles

Conversion between units relies on known equivalences:

• 1 mile = 63,360 inches
• 1 hour = 3,600 seconds

Using these, the conversion process will involve multiplying by ratio factors to cancel out the original units and introduce the desired units.

Step-by-Step Setup of the Conversion

Let’s analyze Tommy’s setup:

1. Starting value: 68 miles/hour
2. Conversion factors:

• Miles to inches: 1 mile = 63,360 inches
• Hours to seconds: 1 hour = 3,600 seconds

1. Conversion process:

• First, convert miles to inches:

\[ 68 \text{ miles/hour} \times \frac{63,360 \text{ inches}}{1 \text{ mile}} \]

• Then, convert hours to seconds:

\[ \frac{1 \text{ hour}}{3,600 \text{ seconds}} \]

• Combine these:

\[ 68 \times \frac{63,360}{1} \times \frac{1}{3,600} \text{ inches/second} \]

Complete setup:

\[
\boxed{
68 \times \frac{63,360}{1} \times \frac{1}{3,600} \quad \text{inches per second}
}
\]

This setup ensures that miles and hours are canceled out, leaving inches per second as the remaining unit.

Verifying the Correctness of Tommy’s Setup

To determine if Tommy’s setup is correct, we analyze each step:

• Are the conversion factors correct?

Yes. 1 mile equals 63,360 inches, and 1 hour equals 3,600 seconds.

• Are the units arranged properly?

Yes. The miles cancel out, leaving inches; hours cancel out, leaving seconds.

• Is the multiplication order appropriate?

Multiplying by inches per mile and dividing by seconds per hour is correct because it aligns with dimensional analysis principles.

• Does the calculation simplify correctly?

Yes. The order of multiplication and division ensures proper unit cancellation and conversion.

Calculating the value:

\[
68 \times 63,360 \div 3,600 = \text{speed in inches per second}
\]

Calculating:

\[
68 \times 63,360 = 4,310,880
\]

Dividing:

\[
4,310,880 \div 3,600 \approx 1,197.47
\]

Therefore, Tommy’s conversion yields approximately 1,197.47 inches per second.

Is Tommy’s Setup Correct? Conclusion

Based on the analysis:

• The units are correctly identified.
• The conversion factors are accurate and appropriate.
• The setup properly cancels out the original units.
• The arithmetic steps are logically sound.

Thus, Tommy’s conversion setup is correct and correctly set up.

Additional Insights and Tips for Accurate Conversions

While Tommy’s setup is correct, here are some tips to ensure accuracy in similar conversions:

• Always verify the conversion factors before applying them.
• Write out units explicitly to track cancellations and avoid errors.
• Perform calculations step-by-step rather than combining everything at once.
• Use a calculator for large numbers to minimize arithmetic mistakes.
• Double-check the final units to confirm they match the desired measurement.

Example Practice Problems

To reinforce understanding, consider practicing similar conversions:

• Convert 55 miles per hour to feet per second. (Use 1 mile = 5,280 feet)
• Convert 100 kilometers per hour to meters per second. (Use 1 km = 1,000 meters)
• Convert 150 miles per hour to yards per second. (Use 1 mile = 1,760 yards)

Applying the same principles and verification steps will help solidify your understanding of unit conversions.

Summary

Tommy’s goal was to convert 68 miles per hour into inches per second, a process requiring careful application of conversion factors and dimensional analysis. His setup, involving multiplying by the number of inches in a mile and dividing by the number of seconds in an hour, is both logically and mathematically correct. Proper setup, verification, and calculation ensure accurate unit conversions, which are essential skills for scientists, engineers, and students alike.

Final Remarks

Understanding unit conversions is more than just plugging numbers into formulas; it involves a clear grasp of units, their relationships, and the correct use of conversion factors. With practice, setting up such conversions becomes intuitive, making problem-solving more efficient and accurate. Tommy’s example demonstrates a correct approach, reinforcing best practices in unit conversions.