The Kentucky Derby is held at the Churchill Downs track in Louisville, Kentucky.
ID: 1275306 • Letter: T
Question
The Kentucky Derby is held at the Churchill Downs track in Louisville, Kentucky. The track is one and one-quarter miles in length. One of the most famous horses to win this event was Secretariat. In 1973 he set a Derby record that would be hard to beat. His average acceleration during the last four quarter-miles of the race was +0.0105 m/s2. His velocity at the start of the final mile (x = +1609 m) was about 16.58 m/s. The acceleration, although small, was very important to his victory. To assess its effect, determine the difference between the time he would have taken to run the final mile at a constant velocity of +16.58 m/s and the time he actually took. Although the track is oval in shape, assume it is straight for the purpose of this problem.
Explanation / Answer
Working formulas:
With acceleration, S = VT + (1/2)aT^2
W/O acceleration, S = VT
where
S = 1609 m
V = 16.58
T = time to negotiate the last mile
a = 0.0105
Substituting values (with acceleration)
1609 = 16.58T + (1/2)(0.0105)T^2
Rearranging the above,
0.00525T^2 + 16.58T - 1609 = 0
Using the quadratic formula,
T = 94.233
Without any acceleration,
1609 = 16.58T
and solving for T,
T = 97.045 sec.
Difference in time = 97.045 - 94.233 = 2.812 sec.
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