Problem 42 - Physics for Scientists and Engineers with Modern Physics

Question

American football uses a field that is 100 yd long, whereas a regulation soccer field is \(100 \mathrm{~m}\) long. Which field is longer, and by how much (give yards, meters, and percent)?

Step-by-Step Solution

Answer

The soccer field is longer by 9.361 yards, 8.565 meters, or 9.361%.

Step 1: Understanding Units and Conversions

The football field is measured in yards while the soccer field is measured in meters. To compare them, we will convert one of these into the other's unit of measurement. We'll convert the 100 meters of the soccer field into yards since we're more used to yards for football fields.

Step 2: Converting Meters to Yards

We know that 1 meter is approximately equal to 1.09361 yards. Thus, to convert 100 meters to yards, we'll multiply by this conversion factor: \[ 100 \,\text{meters} \times 1.09361 \, \text{yards/meter} = 109.361 \, \text{yards} \]

Step 3: Comparing Field Lengths

The soccer field converts to approximately 109.361 yards while the football field is exactly 100 yards. As 109.361 is greater than 100, the soccer field is the longer of the two.

Step 4: Calculating the Difference in Yards

Now we'll calculate the difference in lengths in yards:\[ 109.361 \, \text{yards} - 100 \, \text{yards} = 9.361 \, \text{yards} \]

Step 5: Calculating the Difference in Meters

Since we originally calculated the soccer field's length in meters, the difference is:\[ 100 \, \text{meters} - \left( 100 \, \text{yards} \div 1.09361 \right) = 8.565 \, \text{meters} \]

Step 6: Calculating Percent Difference

To find the percent difference when compared to the original 100-yard football field, use the formula:\[ \left( \frac{9.361}{100} \right) \times 100\% = 9.361\% \]

Key Concepts

Yard to Meter Conversion

Converting units is essential when comparing different measurements. If you have two measurements in different units, you must convert one to compare them effectively. For instance, a soccer field is measured in meters, while an American football field is in yards. To compare these fields, we need to convert the measurements to the same unit.

To convert meters to yards, use the conversion factor that 1 meter is approximately 1.09361 yards. Hence, to find the equivalent yardage of a soccer field that is 100 meters, multiply the length in meters by the conversion factor:

• 100 meters multiplied by 1.09361 yards/meter gives us 109.361 yards.

Now, we have both field lengths in yards, facilitating easy comparison.

Field Length Comparison

Once both fields are measured in the same unit, we can easily compare their lengths. In this example, the 100-yard long football field and the 109.361-yard soccer field can now be directly compared.

From this, it is clear that the soccer field, measured as 109.361 yards, is longer than the football field, which measures 100 yards. This simply shows the importance of converting measurements into common units to understand and compare data effectively. By doing this, we can quickly tell which field occupies more space.

Percentage Difference Calculation

Calculating the percentage difference between two values is useful to understand the proportional difference. In our field length case, we determine how much longer the soccer field is compared to the football field.

To calculate this, use the formula for percentage difference, which is:

• Subtract the two values to find the difference, which is the absolute increase or decrease.
• Divide this difference by the original value (here, the football field's length).
• Multiply by 100% to convert to a percentage.

For example, \[\left( \frac{9.361}{100} \right) \times 100\% = 9.361\% \]This calculation shows that the soccer field is approximately 9.361% longer than the football field, offering a clear picture of their proportional difference.