Question

The current world record for the \(100-\mathrm{m}\) dash is \(9.58 \mathrm{sec}-\) onds held by Usain Bolt of Jamaica. Calculate whether he could have been arrested for exceeding a \(25 \mathrm{mi} / \mathrm{h}\) speed limit while setting that record.

Short answer

Usain Bolt did not exceed the 25 mph speed limit.

Step-by-step solution

Convert the World Record Time to Speed

To find out if Usain Bolt exceeded the speed limit, we first need to calculate his speed in miles per hour. The record time is 9.58 seconds for 100 meters. We need to first convert 100 meters to miles. We know that 1 mile is approximately 1609.34 meters, so Usain Bolt ran: \[ \text{distance in miles} = \frac{100}{1609.34} \approx 0.0621 \text{ miles} \]Next, convert his running time from seconds to hours: \[ \text{time in hours} = \frac{9.58}{3600} \approx 0.002661 \text{ hours} \]Now, calculate his speed: \[ \text{speed in miles per hour} = \frac{0.0621}{0.002661} \approx 23.34 \text{ mph} \]

Compare Bolt's Speed to the Speed Limit

We calculated Usain Bolt's speed to be approximately 23.34 mph. We compare this to the 25 mph speed limit: - Bolt's speed: approximately 23.34 mph - Speed limit: 25 mph Since 23.34 mph is less than 25 mph, he did not exceed the speed limit.

Key concepts

Unit Conversion

Unit conversion is an essential skill in physics as it allows us to express measurements in different units, ensuring that we're using compatible units when performing calculations. In this exercise, we needed to convert meters to miles and seconds to hours.
- **Meters to Miles:** - It's given that 1 mile is equal to 1609.34 meters. - So, to convert 100 meters to miles, you divide 100 by 1609.34. - This is mathematically written as: \[ \text{distance in miles} = \frac{100}{1609.34} \approx 0.0621 \text{ miles} \]
- **Seconds to Hours:** - Since there are 3600 seconds in an hour, convert seconds to hours by dividing the number of seconds by 3600. - For Bolt's 9.58 seconds run: \[ \text{time in hours} = \frac{9.58}{3600} \approx 0.002661 \text{ hours} \]Having the correct units is crucial for accurate calculations and understanding what each number in a problem represents. This conversion enables us to correctly calculate speed in miles per hour, facilitating further analysis.

Speed Calculation

The concept of speed is fundamental in physics as it tells us how fast an object travels over a specified distance in a given amount of time. To calculate speed, we use the formula:
\[ \text{speed} = \frac{\text{distance}}{\text{time}} \]
In our example:- We have already converted Usain Bolt's running distance to miles and his running time to hours.- By plugging these values into the speed formula, we calculate: \[ \text{speed} = \frac{0.0621 \text{ miles}}{0.002661 \text{ hours}} \approx 23.34 \text{ mph} \]Understanding this calculation helps us compare this speed to real-world speed limits. Knowing the conversion factors and calculating speed accurately lets you analyze and solve real-life physics problems.

Comparison Analysis

Comparison analysis involves examining two or more quantities to understand their relationship. In this context, we use it to compare Bolt's calculated speed to a given speed limit.
- **Bolt's Speed:** - After calculations, Bolt's speed is approximately 23.34 mph. - **Speed Limit:** - The speed limit provided is 25 mph. Comparison is straightforward:

• 23.34 mph is less than 25 mph.
• Therefore, Bolt did not exceed the speed limit.

Through comparison, you can draw meaningful conclusions, such as determining whether a speed exceeds a limit. This kind of analysis is widely used in many fields, from daily life scenarios, like speed limits, to more complex scientific studies.