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24 December, 18:46

Two horses leave the starting gate at the beginning of a 2000 m long race. The brown horse has an acceleration of 2.2 m/s2 and a top constant speed of 18 m/s. The black horse has an acceleration of 1.7 m/s2 and a top constant speed of 20 m/s. Which horse wins the race? Assume that each horse's top constant speed is what it can sustain for the rest of the race after the initial acceleration period.

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  1. 24 December, 18:52
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    Define brown horse:

    a₁ = 2.2 m/s², acceleration

    v₁ = 18 m/s, top speed

    Define the black horse:

    a₂ = 1.7 m/s², accelerastopn

    v₂ = 20 m/s, maximum speed

    To travel 2000m, each horse accelerates to maximum speed, and cruises to the finish line.

    For the brown horse:

    The time to attain maximum speed is

    t₁₁ = v₁/a₁ = 18/2.2 = 8.182 s

    The distance travelled during the acceleration is

    s = (1/2) * a*t² = 0.5*2.2*8.182² = 73.636 m

    The time to travel the remaining distance is

    t₁₂ = (2000 - 73.636) / 18 = 107.02 s

    The total time of travel is

    t₁ = t₁₁ + t₁₂ = 8.182 + 107.02 = 115.2 s (approx)

    For the black horse:

    Time to attain maximum speed is

    t₂₁ = 20/1.7 = 11.765 s

    Distance traveled while accelerating is

    0.5 * (1.7 m/s²) * (11.765 s) ² = 117.647 m

    Time to travel the remaining distance is

    t₂₂ = (2000 - 117.647) / 20 = 94.118 s

    The total time of travel is

    t₂ = t₂₁ + t₂₂ = 11.765 + 94.118 = 105.9 s (approx)

    Conclusion:

    The black horse wins the race in about 106 s, while the brown horse takes about 115 s

    Answer: The black horse wins.
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