I'm looking for a simple and or profound answer to my question. If you propel a kilogram to 10 meters per second, you have an energy of 50 joules. If you triple that force impulse and propel a kilogram to 30 meters per second you, have an energy of 450 joules. Even if you made the final momentum of the two objects equal by propelling 3 kilograms to 10 meters per second, you would still only have 150 joules compared to the 450 joules of the kilogram propelled to 30 meters per second. I'm definitely thinking that if you wanted to apply a force that would then be used to do work you would definitely want to get a smaller object moving very fast, than a larger object moving slow. Why is that? Why does force and momentum have a relationship like that with energy, with work?
1 Answers
We can work out the time taken by using the fact that impulse, i.e. force times time, is equal to the change in momentum. So in your example of propelling a mass of 1 kg to 10 m/s the impulse is 10 kg m/s. To propel the same mass to 30 m/s is an impulse of 30 kg m/sec. So assuming the force is the same in both cases:
the time taken triples when the final velocity is tripled
But the work done by the force is not proportional to the time. It is proportional to the distance the object moves. And that distance is given by the equation:
$$ s = \tfrac{1}{2} \frac{F}{m} t^2 $$
So the distance moved is proportional to the time squared, and since the time is proportional to the final velocity that means the distance is proportional to the final velocity squared, so:
the distance moved increases by $3^2$ when the final velocity is tripled
And since the work done by the force is proportional to the distance moved that means the work done by the force increases by a factor of nine. And since the kinetic energy of the object comes from the work done by the force that means the kinetic energy increases by a factor of nine.
And this is exactly what you describe because:
$$ 450 = 50 \times 9 $$
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