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