Consult Concept Simulation 2.1 before starting this problem. The Kentucky Derby

Question

Consult Concept Simulation 2.1 before starting this problem. 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 has never been broken. 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 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

Let us assume that he ran the last 1609m at constant velocity:
Time = distance / velocity
Time = 1609/16.58
Time = 97.04 seconds

Now his time taking into account the acceleration:
s = ut + 0.5a*t²
1609 = 16.58t + 0.5* 0.0105 *t²
1609 = 16.58t + 0.00525t²
0.00525t² + 16.58t - 1609 = 0
Solve the quadratic for t:
t = 94.23seconds (ignore other negative root)

The time difference is 97.04 -94.23 = 2.81 seconds faster

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