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.

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

The black horse wins.