Question

Maurice Greene once ran the \(100-\mathrm{m}\) dash in \(9.81 \mathrm{~s}\) (the wind was at his back), and Khalid Khannouchi ran the marathon (26 mi, 385 yd) in 2:05:42. (a) What are their average speeds? (b) If Maurice Greene could maintain his sprint speed during a marathon, how long would it take him to finish?

Short answer

The average speed of Maurice Greene is approximately \(10.19 m/s\) and the average speed of Khalid Khannouchi is approximately \(5.59 m/s\). If Maurice Greene could maintain his sprint speed during a marathon, it would take him approximately \(1:08:47\) to finish.

Step-by-step solution

Calculation of average speed

Start by calculating the average speed which is defined as the total distance divided by the total time it takes to travel that distance. Use the given distances and times to calculate the average speeds for Maurice Green and Khalid Khannouchi. The distance for Maurice Green's run is \(100 m\) and the time is \(9.81 s\), for Khalid Khannouchi it's \(26 miles, 385 yards\) or approximately \(42,195 m\) and the time is \(2:05:42\) or approximately \(7542 s\). The average speed \(v\) is given by \(v=\frac {d}{t}\), where \(d\) is distance and \(t\) time.

Conversion of units

Convert the distances to meters and times to seconds to ensure consistency while calculating. A mile is about \(1,609 m\) and a yard is about \(0.9144 m\). An hour has 60 minutes and a minute has 60 seconds. Using these conversions, calculate the speed again.

Calculate the time it would take Maurice Greene to finish the marathon if he maintained his sprint speed

Exchange the terms in the formula to calculate the time \(t\) Maurice Greene would need for a marathon, if he could keep up his speed. Given the distance \(d\) and the speed \(v\), the time is calculated by \(t=\frac{d}{v}\). Substitute the sprint speed and the marathon distance into this formula.

Key concepts

Physics Problem Solving

In physics, problem solving involves applying understanding of principles to analyze a given situation. The exercise with Maurice Greene and Khalid Khannouchi showcases how to solve real-life scenarios using physics concepts. The crux of problem solving in this context lies in the precise calculation of average speed, which is a staple concept in kinematics.

To compute average speeds, we first define it as the total distance traveled divided by the time it took to travel that distance. This seemingly simple formula, however, requires careful attention to detail when applying to different situations, as seen between sprinting and marathon running. It's crucial to consider unit consistency and conversion accurately to avoid erroneous results. This application of physics involves not only formulas but also reasoning skills to juxtapose and analyze different scenarios.

Unit Conversion

Unit conversion is a fundamental process in physics problems to ensure that measurements are consistent and comparable. In our exercise, distances and times are given in a mix of units, such as meters, miles, and yards for distance, and seconds and hours for time. To correctly calculate average speed, all units must be in a single standard - typically the metric system.

Steps to Convert Units:

• Understand the conversion factors: 1 mile is approximately 1,609 meters, and 1 yard is 0.9144 meters.
• Time must also be consistent, thus converting hours and minutes to seconds is essential.
• Use these conversion factors to multiply the given values to transit to the desired units.

Mastering these conversions is critical to assure proper calculation of physics problems.

Speed and Distance

Understanding the relationship between speed, distance, and time is key in many physics problems. Speed is defined as the rate at which an object covers a certain distance. If you know any two of the three variables (speed, distance, time), you can solve for the third.

For instance, Maurice Greene's 100m sprint and Khalid Khannouchi's marathon pose a great example of how distance and time affect speed. Short distances typically correlate with higher speeds, as seen with Greene's sprint. Conversely, longer distances, like Khannouchi's marathon, often result in lower average speeds due to physical limitations and endurance factors. Leveraging the formula for average speed, \( v = \frac {d}{t} \), we can gain insights into athletic performance across different sports.

Theoretical Speed Comparison

Theoretical speed comparison involves contrasting calculated speeds in different situations to reason about potential outcomes. For example, the problem questions whether Maurice Greene could maintain his 100m dash speed over a marathon distance.

A theoretical calculation allows us to imagine scenarios outside of typical human performance. By swapping distance with speed in the average speed formula, \( t = \frac{d}{v} \), we can explore hypothetical situations, such as predicting marathon completion times at sprinting speeds. This kind of analysis emphasizes the stark differences in athletic performance across sports, and it underscores the importance of considering realistic physical abilities and limits. Theoretical speed comparisons stretch our understanding of physics by allowing us to calculate and predict outcomes, while simultaneously providing a fun way to engage with real-world data.