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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.

Answer :

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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