paradox in Classical Mechanics? energy vs force

[gusl]

Compare "accelerating under constant power"
with "accelerating under constant force" (constant acceleration)

My intuition tells me that they should be the same, but kinetic energy considerations show that the acceleration is decreasing on the first one (it takes 4 times the energy to get 2 times as fast).

This would seem to contradict "velocity is relative": if velocity were relative, then the energy needed to get faster by 1m/s would be the same whether you are stationary or already at 1 m/s.

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My intuition also tells me that I should be able to come up with a similar paradox about predicting the outcome of a 1-dimensional elastic collision. If you do it with energy vs momentum.

Conservation of energy:
v1_before^2 + v2_before^2 = v1_after^2 + v2_after^2 (if we fix one side of the equation, then the point (v1,v2) falls in a circle)

Conservation of momentum:
v1_before + v2_before = v1_after + v2_after (if we fix one side of the equation, then (v1,v2) falls in a straight line)

The solutions are where circle and line intersect. I guess there's no paradox afterall.

I would like to do a transform to a moving reference frame, to make sure that everything is still alright. Transforming to a fast-moving reference frame will just make the circle bigger. Basically, the point and the line all get transposed diagonally up and to the right. The distance between the intersections still remains the same.

Oh I see, physics is fine. Nothing to worry about.

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The concept of kinetic energy has always been problematic for me. Given the choice, I'll integrate over force instead.

Comments

Re: Energy

[r6.livejournal.com]

I’m not sure I understand the question. Rockets are time agnostic. It doesn’t matter when you decide to throw off your propellent to increase your speed. The only thing that matters is how fast you manage to throw your propellent. The faster you throw your propellent, the more efficient your rocket.

Re: Energy

[gustavolacerda.livejournal.com]

if you shoot two cannonballs at once, will you end up just as fast as if you shot them one by one?

Re: Energy

[r6.livejournal.com]

Yes. The only thing that matters is how fast you shoot them.

Re: Energy

[r6.livejournal.com]

… I think.

Re: Energy

[r6.livejournal.com]

Let’t change that to no.

Re: Energy

[r6.livejournal.com]

If the cannonballs both have an infinitesimal mass, then you will end up just as fast shooting them together as if you shot them one by one. ;-)

Re: Energy

[gustavolacerda.livejournal.com]

I think we could generalize this under the constraint that the total energy spent is the same in both cases.

Suppose you spend E1 when shooting the first cannonball, and E2 shooting the second.
Now go back. If you had shot them both at once, spending E1 + E2, then you should end up with the same speed.

Correct?

Re: Energy

[r6.livejournal.com]

Spending E1 shoots a cannonball at one speed. Spending E2 shoots a cannonball at another speed. In the second case the two cannonballs have the same speed. The two cases are so different, I don’t see why the net momentum of the two cannonballs would be the same in the two cases.

Re: Energy

[gustavolacerda.livejournal.com]

here's another paradox:

From your reasoning before, about pushing oneself off the Earth, it follows that it's more energy-efficient to shoot heavy things: you get more acceleration per rocket this way.

So if you had 2 identical cannonballs and two cannons, then you should shoot them simultaneously, rather than one after the other.

Oh, that makes sense, actually.
...maybe