John and Al are in a 15 km race. John averages 4.4 m/s during the first half of the race and then runs at a speed of 4.2 m/s until the last 200 m, which he covers at 4.5 m/s. At what average speed must Al run to beat John?
The textbook that included the word problem is: Basic Biomechanics Sixth Edition by Susan J. Hall;
ISBN#978-0-07-337644-8
The course name: Biomechanics
I did this one for you yesterday
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> 4.3 m/s
Well, Al, if you want to beat John in this race, you'll have to speed up your running game! Let's see how we can calculate your average speed.
We know that John ran the first half of the race at 4.4 m/s, which means he covered a distance of 7.5 km (half of 15 km).
Then he slowed down a bit to 4.2 m/s for the second half of the race. So, he covered another 7.5 km at this speed.
Finally, he sprinted the last 200 m at 4.5 m/s. We don't need to worry about this part for Al's average speed because it's just a short burst at the end.
Now, let's figure out the time it took for John to complete the race. For the first half, he traveled at a speed of 4.4 m/s for a distance of 7.5 km. So, it took him:
Time = Distance / Speed = 7,500 m / 4.4 m/s = 1,704.55 seconds
For the second half, he traveled at a speed of 4.2 m/s for the same distance of 7.5 km. So, it took him:
Time = Distance / Speed = 7,500 m / 4.2 m/s ≈ 1,785.71 seconds
And for the final 200 m sprint, at 4.5 m/s, it took him:
Time = Distance / Speed = 200 m / 4.5 m/s ≈ 44.44 seconds
Now, let's calculate John's total time for the race:
Total Time = Time for first half + Time for second half + Time for sprint
= 1,704.55 s + 1,785.71 s + 44.44 s
≈ 3,534.70 seconds
So, Al, to beat John, you need to complete the race in less time than 3,534.70 seconds, which is John's total time.
To find your average speed, let's assume you run at a constant speed throughout the race.
Let's call your average speed "A m/s". And since the total distance of the race is 15 km, you traveled the same distance as John.
To calculate your time, we can use the equation:
Time = Distance / Speed
We want your total time to be less than John's time, so we have:
15,000 m / A m/s < 3,534.70 s
Simplifying the inequality, we get:
A > 15,000 m / 3,534.70 s
Calculating this, we get:
A > 4.24 m/s
Therefore, Al, you'll need to run faster than 4.24 m/s on average in order to beat John. Good luck, speedy Al!
To solve the problem, we need to find the average speed at which Al must run to beat John in the race.
First, let's calculate the time it takes for John to complete each part of the race.
For the first half of the race, John averages a speed of 4.4 m/s. The distance of the first half of the race is 15 km / 2 = 7.5 km. To convert this distance to meters, we multiply by 1000: 7.5 km = 7,500 m.
The time taken to cover the first half of the race is:
Time = Distance / Speed = 7,500 m / 4.4 m/s
Next, John runs at a speed of 4.2 m/s for the remaining distance until the last 200 m. The remaining distance is 15 km - 7.5 km - 0.2 km = 7.3 km. Converting this distance to meters, we get: 7.3 km = 7,300 m.
The time taken to cover the remaining distance is:
Time = Distance / Speed = 7,300 m / 4.2 m/s
Lastly, John covers the last 200 m at a speed of 4.5 m/s.
To find the total time taken by John, we add the times for each part of the race.
Total Time = Time for First Half + Time for Remaining Distance + Time for Last 200 m
Now, we can calculate the average speed at which Al must run to beat John.
We know that the total distance of the race is 15 km.
To find Al's average speed, we divide the total distance by the total time taken by John:
Average Speed for Al = Total Distance / Total Time
By solving this equation, we can determine the average speed at which Al must run to beat John in the race.