Question

The 2007 world record for the men's 100 -m dash was \(9.77 \mathrm{~s}\). The third-place runner crossed the finish line in \(10.07 \mathrm{~s}\). When the winner crossed the finish line, how far was the third-place runner behind him? a) Compute an answer that assumes that each runner ran at his average speed for the entire race. b) Compute another answer that uses the result of Example 2.3, that a world- class sprinter runs at a speed of \(12 \mathrm{~m} / \mathrm{s}\) after an initial acceleration phase. If both runners in this race reach this speed, how far behind is the third-place runner when the winner finishes?

Short answer

Also, calculate the distance behind, considering the speed of a world-class sprinter after the initial acceleration phase is 12 m/s. Answer: To find the distance behind, we first calculated the average speeds for both runners and then used these speeds to find the distance covered by the third-place runner when the winner finished. We then subtracted this distance from the total distance to find the distance behind. In the second part of the problem, we used the given world-class sprinter speed of 12 m/s and found the time difference between both runners. We then multiplied the time difference by the sprinter's speed to find the distance behind.

Step-by-step solution

Finding average speeds

For both runners, we will calculate their average speeds throughout the race. The formula for average speed is: Average Speed = Distance / Time For the winner (1st position): Average Speed\(_1 = 100 \mathrm{~m} / 9.77 \mathrm{~s} \) For the third-place runner (3rd position): Average Speed\(_3 = 100 \mathrm{~m} / 10.07 \mathrm{~s} \)

Calculating distance behind for average speeds

Now that we have the average speeds, we can find how far behind the third-place runner was when the winner finished. Since speed = distance / time, we can calculate the distance 3rd place runner ran in the time it took the winner to finish: Distance\(_3 = \)Average Speed\(_3 *\)Winner's Time Then, we can find the distance behind by subtracting the distance run by the third-place runner from the total distance of 100 meters. Distance behind = 100m - Distance\(_3\)

Using the given world-class sprinter speed

In the second part of the exercise, we are given the speed of a world-class sprinter after the initial acceleration phase, which is \(12 \mathrm{~m}/\mathrm{s}\). We will assume both the winner and the third-place runner reach this speed and maintain it for the rest of the race. The problem now becomes finding out how far behind the third-place runner was when the winner finished, given both of their speeds were \(12 \mathrm{~m}/\mathrm{s}\). Since both runners would be running at the same speed, the difference in time it takes for the first and third-place runners to finish the race becomes the essential factor. The time difference can be calculated as: Time difference = Third-place runner's time - Winner's time

Calculating distance behind for the world-class sprinter speed

We will now use the time difference calculated in the previous step to find the distance behind when the winner finished. We can use the formula: Distance behind = World-class sprinter speed * Time difference After calculating the distance behind using this method, we will have our second answer.

Key concepts

Average Speed

Understanding the concept of average speed is essential in kinematics, the branch of physics that deals with the motion of objects. It is simply the total distance traveled divided by the total time taken to cover that distance. To put it in an equation, we express average speed (\t V_avg \t) as: \[\begin{equation}\tV_avg = \frac{distance}{time}\text[\].For instance, when analyzing the performance of sprinters in a 100-meter dash, we calculate their average speed by dividing the 100 meters by the time it took them to complete the race. This gives us an indication of their overall performance throughout the course, not just at a specific point.

In the problem provided, we compared the average speeds of two runners to determine how much distance separates them at the end of the race. This calculation assumes a consistent pace from start to finish, which in reality is an oversimplification, especially considering the dynamic acceleration phase of sprinters, but it provides a useful comparative metric for their general speed over the distance.

Distance Calculation

Distance calculation in kinematics is central to understanding the motion of objects. It involves finding out how far an object has moved over a period of time. In our exercise concerning sprinters, we used the formula for distance based on average speed:\[\begin{equation}\tDistance = Average Speed X Time\text[\],which requires knowledge of the time and the average speed of the object. By calculating how far the third-place runner was able to sprint in the time it took the winner to finish, we were able to deduce the distance behind the winner the third-place sprinter was.

To improve this exercise, focusing on the details of the acceleration phase and the transition to uniform velocity motion would provide a more realistic scenario, reflecting how sprinters actually cover the distance in a race.

Acceleration Phase of Sprinters

The acceleration phase of sprinters is a critical concept in kinematics, especially in athletics. During this phase, sprinters rapidly increase their speed from a stationary position to reach their maximum or near-maximum speed. Sprinters don't maintain a steady pace throughout a race; they actually go through an intense process of accelerating to their top speed, which is then sustained for the remainder of the race.

In physical terms, acceleration is the rate of change of velocity. It is calculated as:\[\begin{equation}\tAcceleration = \frac{change in speed}{time}\text[\].This phase plays a pivotal role in determining a sprinter's performance in short-distance races, such as the 100-m dash. A better understanding and improvement of the acceleration phase can lead to significant improvements in a sprinter's time, which is why athletes and coaches pay close attention to optimizing this aspect of their performance.

Uniform Velocity Motion

Uniform velocity motion is a key concept in kinematics where an object moves at a constant speed in a straight line. This means that the velocity (speed in a given direction) remains consistent over time; acceleration is zero because there’s no change in speed or direction. This concept is particularly important when considering the second part of the race for sprinters, after they have completed their acceleration phase.

When applying uniform velocity to our sprinters' problem, we assume both runners have reached and maintain a speed of 12 m/s after their initial acceleration. With both sprinters moving at this constant speed, the only variable affecting the gap at the finish line is the time difference between them. Therefore, we calculate the distance behind based on the time difference and the constant speed. This is a relatively simplistic model of sprinting kinetics but serves to simplify complex motion into more manageable calculations for basic analysis.