Problem 38 - Chemistry: Principles and Reactions

Question

A lap in most tracks that use the English system is \(0.25 \mathrm{mi}\) (English lap). In countries that use the metric system, a metric lap is \(0.50 \mathrm{~km}\). A champion marathon runner covers a mile in about \(5.0 \mathrm{~min}\). How many minutes will the runner take to run an English lap at that speed? A metric lap?

Step-by-Step Solution

Answer

Answer: The runner would take approximately 1.25 minutes to run an English lap and approximately 1.55 minutes to run a metric lap.

Step 1: Calculate the time taken to cover an English lap

Since the runner's speed is given in minutes per mile, and an English lap is 0.25 miles, we can calculate the time taken to cover an English lap by multiplying the distance by the runner's speed: Time taken (English lap) = Distance (English lap) × Runner's Speed Time taken (English lap) = 0.25 mi × 5.0 min/mi

Step 2: Convert the runner's speed to minutes per kilometer

To convert the runner's speed from minutes per mile to minutes per kilometer, we need to use the conversion factor between miles and kilometers: 1 mile ≈ 1.60934 kilometers Runner's speed in min/km = Runner's speed in min/mi × 1 mi/1.60934 km Runner's speed in min/km = 5.0 min/mi × 1 mi/1.60934 km

Step 3: Calculate the time taken to cover a metric lap

Now that we have the runner's speed in minutes per kilometer, we can calculate the time taken to cover a metric lap by multiplying the distance by the runner's speed in min/km: Time taken (Metric lap) = Distance (Metric lap) × Runner's Speed in min/km Time taken (Metric lap) = 0.50 km × Runner's speed in min/km Now, calculating the values in each step: Time taken (English lap) = 0.25 mi × 5.0 min/mi ≈ 1.25 min Runner's speed in min/km = 5.0 min/mi × 1 mi/1.60934 km ≈ 3.10686 min/km Time taken (Metric lap) = 0.50 km × 3.10686 min/km ≈ 1.55343 min So the runner will take approximately 1.25 minutes to run an English lap and approximately 1.55343 minutes to run a metric lap.

Key Concepts

Distance Conversion

Converting distances between different units is a fundamental skill in many areas, including running, where races can be measured in either miles or kilometers. When you're given a distance in one unit and need it in another, you rely on conversion factors.
To convert miles to kilometers, you use the relation:

• 1 mile ≈ 1.60934 kilometers

Similarly, to convert kilometers to miles, you would use:

• 1 kilometer ≈ 0.62137 miles

These values are derived from standard measurement conversions that take into account the lengths of a mile and a kilometer.
For instance, if you need to convert 5 miles into kilometers, you multiply by the conversion factor:\[5 ext{ miles} \times 1.60934 \frac{km}{mi} ≈ 8.0467 ext{ km}\]Understanding and using these conversions ensures accuracy, especially in races and other measurements.

Speed Calculation

Speed calculation is an important concept for runners and athletes, as it determines how quickly a distance can be covered. Speed is typically a ratio of distance covered to the time taken.
In running, it is often expressed in units like minutes per mile or minutes per kilometer.

• To find the speed in a specific unit, you adjust based on the distance and time given.

For example, if a runner covers a mile in 5 minutes, their speed is:\[ ext{Speed} = rac{5 ext{ min}}{1 ext{ mi}} = 5 ext{ min/mi} \]Converting this speed to minutes per kilometer requires understanding of distance conversion, as in our earlier example:

• You adjust the time it takes, using the appropriate conversion factor.

This calculation helps in planning and evaluating performance during events.

Marathon Running

Marathon running is a challenging endurance event that spans a distance of 26.2 miles or approximately 42.195 kilometers.
It tests the limits of a runner's stamina and speed, requiring rigorous preparation and training.
Elite marathon runners often maintain impressive speeds throughout the race, making their timing in any distance significant.

• A common milestone is running a mile under 5 minutes for elite athletes.
• Consistency in pace is crucial, especially around the course's varying topography.

For example, in a high-level marathon, knowing how quickly a runner covers laps or shorter distances helps in assessing overall pace.
Such insights into lap times and conversions are essential for runners looking to optimize their performance.

Metric and Imperial Systems

The metric and imperial systems are two major systems of measurement used worldwide. Each has its own units for measuring length, area, and volume.
The imperial system, used primarily in the United States, includes miles, yards, feet, and inches.
Conversely, the metric system, adopted by most other countries, uses meters, kilometers, centimeters, and millimeters.

• The metric system is known for its ease of conversion using powers of ten.
• The imperial system, while less intuitive for conversion, is deeply ingrained in certain cultures.

For a runner familiar with both, being able to mentally convert between the systems can be highly advantageous for understanding race conditions and distances. Transitioning smoothly between these systems requires practice, which is often achieved through regular measurement in both sets of units.