The Kentucky Derby is held at the Churchill Downs track in Louisville, Kentucky.

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 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

Cool fun fact Q,
first lets do the easy part and figure out how long the mile would take at constant v.
just take the distance 1609m and divide by the velocity 16.58.
1609/16.58=97s

now lets do the time with acceleration. This is a Newton's equations of motion question.
use the equation
x=xi+vi(?t)+1/2*a(?t)^2
lets say xi=0
x=1609
vi=16.58
a=.0105
now solve for ?t lets just call it t for simplicity
need to set up the quadratic equation
0=.00525t^2+16.58t-1609
using the quadratic equation (just google this if you don't know what it is)
we get you get two answers but only one makes sense the 94.2s

so there would have been about a 2.8s difference and in horse racing that is a lot   

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