0

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?

James Montagne
  • 41
  • "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 = mv, so one will have 1kg10m/s and the other 3kg*10m/s. Otherwise your calcs seem correct. –  May 12 '19 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 '19 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. –  May 12 '19 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 '19 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 '19 at 15:56
  • Can't anybody tell me why a change in energy is the square of a change in velocity? – James Montagne May 14 '19 at 15:16
  • You cannot prove or disprove or derive a definition. It sounds like you are not asking how something works but asking the physics equivalent of "define the meaning of is." It does what it does. (1) not all energy relations are quadratic in speed, relativistic kinetic energy is more complicated. (2) The Energy is equal to the work done which is DEFINED as the integral of the applied force over the displacement of the particles path. A little calculus will get you to the result. You are confusion impulse with force. –  May 14 '19 at 15:46
  • Yes, you're absolutely right. I'm looking more for a why is it this way, than a how. I fully understand the how. In the textbook I studied momentum and change in momentum came first, and then they introduced energy and work. Of course I tried to overlay them, and deduced that a change in momentum is the same as a change in velocity, and that the energy, or work, reaped was exponential to them ( impulse aside, I realize it's just the duration of a force). What I wanted what some profound physics explanation as to why the relationship between the two are the way they are. – James Montagne May 14 '19 at 16:58
  • There isn't imo, other than possibly that both can be derived form Newton's law of motion and the definition of work. And, the relation is only approximate as it is not this way in relativity. –  May 14 '19 at 17:53
  • I'm starting to see that. All I've seen on the internet and the exchange is that mass increases with velocity (as you mentioned, as in relativity), but I don't see how that could apply to differences in velocity of tens of meters per second. I don't think something has four times the mass because you double the velocity. Do you have any advice on how to frame this question so that is easier to understand and might get more attention on the exchange? – James Montagne May 15 '19 at 14:32
  • Mass does not increase with velocity. this is an antiquated ad hoc rule for getting the correct factor in the relativistic energy equation. The rest mass is the norm of a 4-vector and hence invariant. –  May 15 '19 at 14:59
  • I kind of figured that, concerning mass. It wouldn't apply to these small numbers that I'm dealing with anyway. I'm just wondering how to phrase the question in the future on the exchange, why a smaller faster mass has more energy even if it has the same momentum as a larger mass object. – James Montagne May 15 '19 at 15:59

1 Answers1

1

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:

$$ 450 = 50 \times 9 $$

John Rennie
  • 355,118
  • 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 '19 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 '19 at 14:46
  • I think this question should be closed. –  May 15 '19 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 '19 at 16:17
  • @John Rennie You've probably looked at my other comments concerning this question, but I don't leave people hanging so I'll answer your question. Essentially yes. With momentum, mass and velocity are proportional, but kinetic energy is the square of velocity. Momentum is good for figuring out what happens in collisions, but energy describes what an object can do, how many other objects that can push around before it is expended. I'm trying to figure out why faster objects can do more work than slower ones even if they're momentum is the same. That's it in a nutshell. – James Montagne May 16 '19 at 15:25
  • 1
    @JamesMontagne see Why does kinetic energy increase quadratically, not linearly, with speed? – John Rennie May 16 '19 at 15:42
  • @John Rennie Exactly. I know that kinetic energy is the force and the distance it takes get an object to a certain velocity. That's not the question, that's established. I simply want to know why is it like that. Why can faster objects do exponentially more than slower objects in the universe. I'm sure that it involves some theory, but maybe not. Also, if you don't have an answer to the question, I would like to know how you would phrase the question on the exchange. I would like to get more positive responses from people. – James Montagne May 17 '19 at 16:24
  • @John Rennie I'm sorry, I didn't realize you were giving me a link to another question. I thought you were asking the question. Thanks for your help. – James Montagne May 18 '19 at 17:32