Problem 49 - Chemistry The Science in Context

Question

Boston Marathon To qualify to run in the 2015 Boston Marathon, a distance of 26.2 miles, an 18 -year-old woman had to have completed another marathon in 3 hours and 34 minutes, or less. To qualify, what must this woman's average speed have been (a) in miles per hour and (b) in meters per second?

Step-by-Step Solution

Answer

Answer: (a) Approximately 7.34 mph and (b) approximately 3.28 m/s.

Step 1: Convert Time to Hours

First, we need to convert the given time of 3 hours and 34 minutes into hours: 3 hours + 34 minutes = 3 hours + (34/60) hours Since there are 60 minutes in an hour, we can divide the number of minutes (34) by 60 to convert it into hours. So, the time in hours is: 3 + (34/60) = 3.5667 hours (rounded to 4 decimal places)

Step 2: Calculate Speed in Miles per Hour

Now we can calculate her average speed in miles per hour (mph) using the formula: Speed (mph) = Distance (miles) / Time (hours) We know the distance she has to cover is 26.2 miles, and we've found her time to be 3.5667 hours: Speed (mph) = 26.2 / 3.5667 ≈ 7.34 mph (rounded to 2 decimal places)

Step 3: Convert Distance to Meters and Speed to Meters per Second

Next, we need to convert the distance she has to cover (26.2 miles) to meters. We know 1 mile = 1609.34 meters (in exact value), so: Distance (meters) = 26.2 miles × 1609.34 meters/mile ≈ 42164.68 meters (rounded to 2 decimal places) Now we'll convert the time in hours to seconds. We know there are 3600 seconds in an hour, so: Time (seconds) = 3.5667 hours × 3600 seconds/hour ≈ 12840.12 seconds (rounded to 2 decimal places) Now we can calculate her average speed in meters per second (m/s) using the formula: Speed (m/s) = Distance (meters) / Time (seconds) Speed (m/s) = 42164.68 / 12840.12 ≈ 3.28 m/s (rounded to 2 decimal places) So, this woman's average speed to qualify for the 2015 Boston Marathon must have been (a) approximately 7.34 miles per hour (mph) and (b) approximately 3.28 meters per second (m/s).

Key Concepts

Average Speed

Understanding average speed is key when it comes to calculating how fast someone or something is moving over a specific distance. Average speed is determined by dividing the total distance traveled by the total time taken to travel that distance.
For example, in the Boston Marathon exercise, the runner needs to figure out her average speed to qualify for the race. The formula to calculate average speed is straightforward:

• Average Speed = Total Distance / Total Time

Remember, average speed is not necessarily the same as the speed at every moment during the race. It provides an overall view of how fast the runner was going on average. Using this, we can get a clear picture of the performance needed to meet marathon qualification standards.

Unit Conversion

Unit conversion is an essential skill, especially when dealing with speed calculations, because speeds can be expressed in various units. Converting between these units ensures that all measurements are in a consistent system, which is necessary for accurate calculations.
In the context of the Boston Marathon problem, the runner's pace is initially provided in miles, but we also need to express it in meters.

• Conversion from miles to meters: 1 mile equals 1609.34 meters.
• Time units may also need conversion; 1 hour equals 3600 seconds.

When needing to convert between different units of speed, remember to account for both the distance and the time conversions. This ensures that your calculations are accurate and applicable in different measurement contexts.

Distance and Time Calculation

Calculating distance and time accurately is fundamental to determining speed. You should always start by ensuring your units are consistent. This means converting all measurements to the same system, such as using hours and meters throughout your calculations.
In our situation, the runner's goal is set in hours and miles. To find her pace in another unit like meters per second, she first converts distance to meters and time to seconds. The steps include:

• Convert the running distance: Multiply mileage by the meters per mile conversion factor.
• Convert the time: Multiply the hours by the seconds in an hour.

These conversions enable the calculation of speed in terms of meters per second. By understanding these steps, you can simplify complex problems into manageable calculations, leading to accurate and meaningful results.