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Other day I got into a chat with my friend on work power and energy where he asked an intelligence related question the question was like this

if a race was conducted between a man a horse and a car of SAME POWER then who will most probably win the race

and to his question i answered that obviously man could win it because the definition of power states that

power is the fraction of work done in a given time

and as per the definition of work it is described as the dot product of force and displacement

and from newton's second law we get F=ma

the displacement of all the three is same as their end point. so the product of mass and acceleration must be same for all and in this situation a normal man will have less weight than horse and car so it clearly states that acceleration must be of more magnitude for man.

but to my surprise he said my answer is wrong and he did'nt explained how it was wrong

am i really wrong? please guide me if i am wrong in which aspect.

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that depends on the length of the race. If it is a sprint, the horse. A 5k, probably the car. A day-long marathon, the man is sure to win. A horse can accelerate faster than a car of the same power by virtue of the spring action of their hind legs, good for the sprint. A car can generally utilize power more efficiently and keep a steady pace, good for the 5k. But humans are built for marathon running; horses don't sweat like us, they will overheat and cars run out of fuel. Humans win in the long marathons –  Jim Jul 4 at 17:29
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If all three contestants were of the same design but the weight of the respective objects (car, horse, man). Then the lightest object would always win. However, one needs to consider the mechanisms in play. A horse and a man of the same power do not utilize that power the same way or equally efficiently –  Jim Jul 4 at 17:38
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2 Answers

up vote 2 down vote accepted

Let's work through this problem...

$$\text{Power} = \frac{\text{Work}}{\text{Time}} \; \text{and} \; \text{Work} = \text{Force}\cdot \text{Displacement} $$

therefore

$$\text{Power} = \frac{\text{Force}\cdot\text{Displacement}}{\text{Time}} $$

From Newton's Second law we know

$$\text{Force} = \text{Mass}\cdot\text{Acceleration}$$

Substituting again:

$$\text{Power} = \frac{\text{Mass}\cdot\text{Acceleration}\cdot\text{Displacement}}{\text{Time}}$$

Now the only things that will differ in your set up is the mass (since you want to know the greatest acceleration)

$$ \text{Mass} = \frac{\text{Power}\cdot\text{Time}}{\text{Acceleration}\cdot\text{Displacement}}$$

This relationship implies that, all others held constant, to get a greater acceleration you have a smaller mass. So the Man would win because in the same amount of time he can accelerate to a greater velocity.

You were right and your friend was probably just mad he was wrong :)

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Thanks for the readability edit @ACuriousMind –  jkeuhlen Jul 4 at 16:01
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You're welcome. If you want to write pretty looking TeX on your own, here's a basic tutorial for math.SE that carries over well to physics.SE. –  ACuriousMind Jul 4 at 16:03
    
Awesome thanks! As you can probably tell I'm new :) –  jkeuhlen Jul 4 at 16:05
    
thanks a lot and special thanks for that last line :) –  agha rehan abbas Jul 4 at 16:34
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in this situation a normal man will have less weight than horse and car so it clearly states that acceleration must be same.

No. If the force is the same for all 3, then if the mass is different, the acceleration must be different. Since the man has the least mass, then he will accelerate the fastest. So the man would win the race.

But in the real world, a man cannot develop 1 hp using his legs (I think it is about 0.6 hp). A house would develop 1 hp and a car is usually rated over 80 hp. It would certainly be difficult to say whether a horse or a car would win a short race (a car has a faster top speed so it will win a long race).

A race between horses and men. Another race.

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sorry i mistyped my question and now i have edited it –  agha rehan abbas Jul 4 at 16:17
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