Question

The world record for the hundred meter dash is $9.74\mathrm{s}$. What is the corresponding average speed in units of $\mathrm{m}/\mathrm{s},\mathrm{km}/\mathrm{h},\mathrm{ft}/\mathrm{s}$, and $\mathrm{mi}/\mathrm{h}$ ? At this speed, how long would it take to run $1.00\times {10}^2$ yards?

Short answer

The corresponding average speeds for the world record 100-meter dash time of 9.74 seconds are as follows: 10.27 m/s, 36.97 km/h, 33.69 ft/s, and 22.97 mi/h. At this speed, it would take approximately 8.90 seconds to run 100 yards.

Step-by-step solution

Convert time to seconds

First, let's represent the given time in seconds: $9.74\mathrm{s}$

Calculate average speed in m/s

Now, we can find the average speed in m/s simply by dividing the distance by the time: Average speed = $\frac{Distance}{Time}$ = $\frac{100\mathrm{m}}{9.74\mathrm{s}}$ = $10.27\mathrm{m}/\mathrm{s}$

Convert m/s to km/h

To convert the average speed from m/s to km/h, we can use the following conversion: 1 m/s = 3.6 km/h So, the average speed in km/h = $10.27\mathrm{m}/\mathrm{s}\ast 3.6\frac{\mathrm{km}}{\mathrm{h}}$ = $36.97\mathrm{km}/\mathrm{h}$

Convert m/s to ft/s

To convert the average speed in m/s to ft/s, we can use the following conversion: 1 meter = 3.281 feet So, the average speed in ft/s = $10.27\ast 3.281\mathrm{ft}/\mathrm{s}$ = $33.69\mathrm{ft}/\mathrm{s}$

Convert ft/s to mi/h

To convert the average speed from ft/s to mi/h, we can use the following conversion: 1 ft/s = 0.6818 mi/h So, the average speed in mi/h = $33.69\ast 0.6818\mathrm{mi}/\mathrm{h}$ = $22.97\mathrm{mi}/\mathrm{h}$

Convert the distance to yards

Next, we need to convert the given distance of 100 yards to meters: 1 yard = 0.9144 meters So, the distance in meters = $100\ast 0.9144\mathrm{m}$ = $91.44\mathrm{m}$

Calculate time to run 100 yards

Now, we can use the average speed in m/s and the distance in meters to calculate the time required to run 100 yards at the world record speed: Time = $\frac{Distance}{Speed}$ = $\frac{91.44\mathrm{m}}{10.27\mathrm{m}/\mathrm{s}}$ = $8.90\mathrm{s}$ Thus, it would take approximately 8.90 seconds to run 100 yards at the world record speed.

Key concepts

Speed Conversion

Understanding speed conversion is crucial when dealing with exercises that require reporting the result in various units. This concept refers to the process of changing the value of speed from one unit of measurement to another without altering the actual pace. For example, converting the average speed of a runner from meters per second (m/s) to miles per hour (mi/h) involves multiplying or dividing by specific conversion factors, such as using 3.6 to go from m/s to kilometers per hour (km/h).

An important step is to know the most common conversion factors or to have access to a conversion chart. Keep in mind the unit congruence; units on both sides of an equation must be consistent. In some cases, multiple conversions might be necessary to get the final desired unit—each conversion being a simple multiplication or division operation. Practice with various speeds and units strengthens one's ability to quickly and accurately perform these conversions.

Unit Conversion

Unit conversion is absolutely necessary not only in physics but also in many real-world applications. It involves calculating the equivalent amount of one measurement unit in terms of another unit of the same dimension. For example, when converting feet to meters or yards to meters, as seen in the exercise, the key lies in knowing the precise conversion factor between the units. In the provided solution, using 0.9144 meters for each yard was essential to ascertain the distance in a unit compatible with the speed previously calculated in meters per second.

Since units such as distance and time are fundamentals in speed calculations, effective problem-solving requires students to master this skill. Utilizing the correct conversion factors lets us accurately translate one measurement into another, securing the integrity of computations and conclusions derived from them.

Problem-solving in Physics

Physics problems often involve a systematic approach to finding a solution. The process begins with understanding the question, identifying what is given and what needs to be found—a fundamental step to avoid confusion during calculations. Next involves formingulating a strategy, such as using the formula for average speed, which is the distance traveled divided by the time taken. In this example, the initial task was to determine the average speed in various units given the world record for a distance in meters and time in seconds.

In physics, problem-solving requires careful reasoning and logical steps, verified with unit consistency and conversion where necessary. Knowing to work sequentially, like how it was first necessary to calculate the speed in m/s before converting it to other units, is key. Applying these problem-solving techniques allows for a clear path to an accurate and comprehensive solution in physics exercises.