Problem 20 - Physics for Scientists and Engineers with Modern Physics

Question

(II) How much longer (percentage) is a one-mile race than a 1500 -m race ("the metric mile")?

Step-by-Step Solution

Answer

The one-mile race is approximately 7.29% longer than the 1500-meter race.

Step 1: Convert Mile to Meters

First, we need to convert the one-mile distance to meters so that we can compare it with the 1500 meters directly. Since 1 mile is equivalent to 1609.34 meters, we will use this conversion for our calculations.

Step 2: Calculate the Difference in Length

Next, we find the difference in length between the one-mile race and the 1500-meter race. We do this by subtracting the length of the 1500-meter race from the converted length of the one-mile race.\[\text{Difference} = 1609.34 \text{ m} - 1500 \text{ m} = 109.34 \text{ m}\]

Step 3: Find the Percentage Increase

To find out how much longer the one-mile race is compared to the 1500-meter race, we calculate the percentage increase. The formula for percentage increase is:\[\text{Percentage Increase} = \left( \frac{\text{Difference}}{\text{Original Length}} \right) \times 100\]Plug in the values:\[\text{Percentage Increase} = \left( \frac{109.34}{1500} \right) \times 100 \approx 7.29\%\]

Step 4: Conclusion

The one-mile race is approximately 7.29% longer than the 1500-meter race.

Key Concepts

Distance Conversion

Distance conversion is an essential skill in mathematics and science, especially when dealing with different unit systems. To convert between metric and imperial units, such as converting miles to meters, you need to know the conversion factor. For example:

• 1 mile equals 1609.34 meters.

To convert miles to meters, multiply the number of miles by 1609.34. This allows you to compare or calculate distances in different units. Here, converting 1 mile to meters helps understand how a mile measures up against the metric-based 1500 meters in a race.
This simple skill is useful in real-world applications such as athletics and travel planning.

Length Measurement

Understanding length measurement helps compare and analyze distances effectively. Length can be measured in various units, such as meters, kilometers, miles, and inches. The choice of units often depends on the context of the activity or analysis.
In athletics, races are often measured in both meters and miles, but metric units (like meters) are common as they provide a standard reference.
To determine how much longer one distance is compared to another, subtract the shorter distance from the longer one.

• For example, finding the difference between 1609.34 meters and 1500 meters:
• 109.34 meters.

This simple calculation helps to visualize the difference between two lengths.

Percentage Increase Calculation

Percentage increase calculation is a handy tool to understand how much larger one value is compared to another as a percentage. To calculate the percentage increase:
Use the formula:

• \( \text{Percentage Increase} = \left( \frac{\text{Difference}}{\text{Original Length}} \right) \times 100 \)

In this scenario, the difference in distance is 109.34 meters, and the original length is 1500 meters. Divide the difference by the original length, then multiply by 100 to get a percentage. This method provides a clear picture of the difference relative to the original measure, which is particularly useful when comparing the lengths of races or other distances.