# Why is energy exponential with respect to the force impulses required to achieve those energies? [closed]

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?

• "Even if you made the final momentum of the two objects equal by propelling 3 kilograms to 10 meters per second,". This will not make the final momentum the same in these cases. p = m*v, so one will have 1kg*10m/s and the other 3kg*10m/s. Otherwise your calcs seem correct. – ggcg May 12 at 15:35
• No, I meant that the 3 kg accelerated to 10 m per second would be equal in momentum to the 1kg accelerated to 30 meters per second example. And given that they received the same force and they have the same momentum, and that the 3 kg would have an energy of 150 joules and the 1 kilogram would have an energy of 450 joules, there is a big difference between the force impulse exerted on both masses and the energies they possess after the acceleration. If you don't understand the nature of the question, I don't know that you're the person that would be able to answer it. – James Montagne May 12 at 17:49
• You question was poorly worded. Perhaps you should consider rewording it if you want serious answers. Also, equal impulse does not equate to equal force, they have different units. Just because dp is the same does not mean the F(t) or dt for the processes are the same. – ggcg May 12 at 18:04
• Maybe, but I was just trying to get across the gist of the question by using examples. The gist of it is if you apply the same force impulse to two different mass objects, the smaller one will have more energy and capacity to do work. And, the bigger the difference in mass of the two objects the bigger the difference in energy as well. That's what I want to know. Why is that? – James Montagne May 13 at 15:34
• Or, if you have two objects the same mass and you apply different size impulses to them, the energies of those objects is going to be exponential in relation to the difference of the size of the impulses. Same phenomena, just different examples. I want to know why the phenomenon exists and what causes it. – James Montagne May 13 at 15:56

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:

$$450 = 50 \times 9$$

• I know my math and assertions are correct, I've given it a lot of thought. What I want to know is why. If you double the force on an object you double its momentum, but you quadruple its energy. If you increase the force of an object by 10, you increase the momentum by 10, but you increase the energy 100 fold. I realize that, that's not what I'm asking. What I want to know is why. – James Montagne May 15 at 14:40
• @JamesMontagne I confess I'm uncertain what exactly you are asking. Are you asking why KE is proportional to $v^2$ while momentum is proportional to $v$? – John Rennie May 15 at 14:46
• I think this question should be closed. – ggcg May 15 at 15:00
• I think you're right. I'm looking for a preponderance of the data, and the question is probably just going to get people throwing the data around that I'm already aware of. – James Montagne May 15 at 16:17
• – John Rennie May 16 at 15:42