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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. Number Units

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Answer:

2.815 seconds

Explanation:

t = Time taken

u = Initial velocity

v = Final velocity

s = Displacement

a = Acceleration

[tex]s=ut+\frac{1}{2}at^2\\\Rightarrow 1609.34=16.58\times t+\frac{1}{2}\times 0.0105\times t^2\\\Rightarrow \\\Rightarrow 4.905t^2+16.58t-1609.34=0[/tex]

[tex]t=\frac{-16.58+\sqrt{308.69254}}{0.0105},\:t=\frac{-16.58-\sqrt{308.69254}}{0.0105}\\\Rightarrow t=94.25, -3252\ s[/tex]

Time taken with the acceleration is 94.25 seconds

Time = Distance / Speed

[tex]\text{Time}=\frac{1609.34}{16.58}\\\Rightarrow \text{Time}=97.065\ s[/tex]

Difference in time = 97.065-94.25 = 2.815 seconds

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