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3 years ago

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A man's higher initial acceleration means that a man can outrun a horse over a very short race. A simple - but plausible - model

for a sprint by a man and a horse uses the following assumptions: The man accelerates at 6.0 m/s2for 1.8 s and then runs at a constant speed. A horse accelerates at a more modest 5.0 m/s2 but continues accelerating for 4.8 s and then continues at a constant speed. A man and a horse are competing in a 200 m race. The man is given a 100 m head start, so he begins 100 mfrom the finish line.
How much time does it take the man to finish the race? The horse? Who wins?

1 answer:

Lera25 [3.4K]3 years ago

5 0

Answer:

t_man = 10.16 s, t_horse = 10.73 s, the winner is the man

Explanation:

To solve this problem we are going to find the time of each one separately.

Man we look for distance and time during acceleration

x₁ = v₀ t₁ + ½ a₁ t₁²

as it comes out of rest its initial velocity is zero

x₁ = ½ a₁ t₁²

x₁ = ½ 6.0 1.8²

x₁ = 9.72 m

at this point its speed is

v₁ = v₀ + a t

v₁ = 0 + 6 1.8

v₁ = 10.8 m / s

From here on it continues at constant speed, the distance that the man needs to travel from the point where the man leaves at 100m is

x₂ = 100 - x₁

x₂ = 100- 9.72

x₂ = 90.28 m

the time for this part is

v₁ = x₂ / t₂

t₂ = x₂ / v₂

t₂ = 90.28 / 10.8

t₂ = 8.36 s

the total time for the man is

t_man = t₁ + t₂

t _man = 1.8 + 8.36

t_man = 10.16 s

We repeat the calculation for the horse

distance traveled during the acceleration period

x₃ = v₀ t + ½ a₂ t₃²

as part of rest its initial velocity is zero

x₃ = ½ a₂ t₃²

x₃ = ½ 5.0 4.8²

x₃ = 57.6 m

the velocity at this point is

v₃ = v₀ + a₂ t₃

v₃ = 0 + 5 4.8

v₃ = 24 m / s

the rest of the route is at constant speed, the remaining distance

x₄ = 200 - x₃

x₄ = 200 - 57.6

x₄ = 142.4 m

the time to go through it is

t₄ = x₄ / v₃

t₄ = 142.4 / 24

t₄ = 5.93 s

the total time for the horse is

t_horse = t₃ + t₄

t_horse = 4.8 + 5.93

t_horse = 10.73 s

when we compare the times we see that the man arrives a little before the horse, the winner is the man

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