Secretariat ran the Kentucky Derby with times for the quarter mile of 25.2 s, 24.0 s, 23.8 s, and 23.0 s. (a) Find his average speed during each quarter-mile segment. (b) Assuming that Secretariat's instantaneous speed at the finish line was the same as the average speed during the final quarter mile, find his average acceleration for the entire race. (Hint: Recall that horses in the Derby starts from rest.)
a) Divide 0.25 miles by the time for each quarter. That will give you the average speed for each quarter in miles/sec. Multiply that by 5280 for feet/sec.
b) Divide the finish line (last quarter average) speed by the total race time for the average acceleration.
Whoever dreamed up this question apparently did not realize that the Kentucky Derby distance is 1 1/4 miles, not one mile. Someone left out the time of one of the quarter miles. The answer you get assuming this to be a mile race will be wrong.
To find Secretariat's average speed during each quarter-mile segment, you need to divide the distance traveled in each segment by the time taken.
(a) Average speed during each quarter-mile segment:
1. The distance traveled in the first quarter-mile segment is 1/4 mile.
Average speed = Distance / Time = 1/4 mile / 25.2 s.
2. The distance traveled in the second quarter-mile segment is also 1/4 mile.
Average speed = Distance / Time = 1/4 mile / 24.0 s.
3. The distance traveled in the third quarter-mile segment is again 1/4 mile.
Average speed = Distance / Time = 1/4 mile / 23.8 s.
4. The distance traveled in the fourth quarter-mile segment is 1/4 mile.
Average speed = Distance / Time = 1/4 mile / 23.0 s.
(b) To find Secretariat's average acceleration for the entire race, we need the time taken for the final quarter mile.
1. Calculate the total time for the three quarters:
Total time = 25.2 s + 24.0 s + 23.8 s = 72.0 s.
2. Calculate the time taken for the final quarter:
Time taken for the final quarter = Total time - Time taken for the first three quarters.
Time taken for the final quarter = 72.0 s - (25.2 s + 24.0 s + 23.8 s).
3. Average acceleration = Change in velocity / Time taken for the final quarter mile.
Since Secretariat's instantaneous speed at the finish line was the same as the average speed during the final quarter mile, the change in velocity is zero.
Therefore, Average acceleration = 0 m/s^2.
So, the average speed during each quarter-mile segment is the distance traveled in that segment divided by the time taken. The average acceleration for the entire race is zero, as the final speed is the same as the average speed during the final quarter mile.
To find Secretariat's average speed during each quarter-mile segment, we need to divide the distance covered in each segment by the time taken.
(a) Average speed for the first quarter-mile:
Distance = 1/4 mile
Time = 25.2 s
Average speed = Distance / Time
Average speed = (1/4) / 25.2
Average speed = 0.01 miles/s
Average speed for the second quarter-mile:
Distance = 1/4 mile
Time = 24.0 s
Average speed = (1/4) / 24.0
Average speed = 0.0104 miles/s
Average speed for the third quarter-mile:
Distance = 1/4 mile
Time = 23.8 s
Average speed = (1/4) / 23.8
Average speed = 0.0105 miles/s
Average speed for the fourth quarter-mile:
Distance = 1/4 mile
Time = 23.0 s
Average speed = (1/4) / 23.0
Average speed = 0.0109 miles/s
(b) Assuming Secretariat's instantaneous speed at the finish line was the same as the average speed during the final quarter-mile, we can calculate his average acceleration for the entire race.
Average acceleration = (Final speed - Initial speed) / Time
The initial speed is 0 miles/s because the horses start from rest.
The final speed is the average speed during the last quarter-mile, which is 0.0109 miles/s.
Time is the total time it takes to complete the race, which is the sum of the times for each quarter-mile.
Total race time = 25.2 + 24.0 + 23.8 + 23.0 = 95 seconds
Average acceleration = (0.0109 - 0) / 95
Average acceleration = 0.0001147 miles/s^2
Therefore, Secretariat's average acceleration for the entire race is approximately 0.0001147 miles/s^2.