Wow is this a blast from the past! I'm the original poster of the question and certainly didn't expect to bump in to it here so many years later!
Why does kinetic energy increase quadratically, not linearly, with speed? (2011)
221–230 of 231 posts
Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)
#222Earlier quoted context omitted.
Because things like energy are relative. So if you label the ground 0, and go up 10 feet, you get x energy. Going up another exact same x from your 10 foot ladder spot you could now call 0 again, would mean you gain x energy again. Since they're both the same height, and you gained the same energy, you could infer double the height has double energy.
What if you label standing still as 0 mph and start moving 10 mph, gaining x energy, then call that zero and start moving 10 mph from there? It's just as intuitive to say that you would gain x energy in that case, but you don't.
And your intuition is actually right!
As a Gedankenexperiment:
If you are on a train that moves 10 mph and you toss a ball to give it another 10 mph of velocity (relative to the train), your arms need to perform the same amount of work on the moving train as on a stationary train.
Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)
#223Earlier quoted context omitted.
I’m not talking about intuitions; I’m talking against them. The intuition about carrying something 1st to 2nd then to 3rd floor is clearly wrong as evidenced by the example I gave; it is less wrong in smaller scales, but it still is wrong. If the floors were as high as the radius of the Earth, the first one would be three times as hard as the second one. The math doesn’t come out the same. It’s not at all linear, it’…
Simplifying assumptions are allowed when reasoning about physics. What you are saying is interesting, but I think you might be misapplying it. For small heights differences, the difference in gravitation approaches 0. A ball raised .0002 mm off the ground has twice the potential energy of one raised only .0001 mm. But that is never true for speed. An object moving 0.0002 MPH does have 4x the kinetic energy of an obje…
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Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)
#224K = (1/2)mv^2 + (3/8)mv^4/c^2 + (5/16)mv^6/c^4 + (35/128)mv^8/c^6 + (63/256)mv^10/c^8 + (231/1024)mv^12/c^10 + ...
Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)
#225It's easiest to visualize in terms of conversion from potential energy. We know intuitively that a ball atop a 20ft ladder has twice the potential energy of a ball atop a 10ft ladder. And we also know when they fall, by the time they reach the ground and all the potential energy has been converted to kinetic energy, the previously higher ball will have twice the kinetic energy too. But a twice higher ball won't have…
I agree that this feels intuitive, that potential energy should increase linearly with height. But in the end, it's all up to the units/quantities we choose to measure, no? If we, say, decided to measure "Squenergy" in Sqoules, with 1Sq² = 1J, then suddenly, squenergy does increase linearly with speed! The formula for kinetic Squenergy becomes sqrt(m/2)v. Of course this complicates other stuff, like potential Squener…
The difference is that Squenergy is not conserved.
Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)
#226Earlier quoted context omitted.
Put that block on a shelf at the same height. 0 work. The fact that your muscles burn ATP just fighting gravity is a feature of biology, not fundamental to the physics involved. If you want to read about another similar example, Google for rocket launch gravity losses.
I think you're missing the point. A lot of basic mechanics actually isn't especially intuitive because things like work simply do not map well to everyday experience. I'm not suggesting that work is defined incorrectly or something.
I see what you're saying now, and I agree.
Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)
#227Earlier quoted context omitted.
> Every time in physics you see quadratic, you should think sphere. Not sure how I reconcile that for systems with linear symmetry that don't admit a sphere such as a 1D harmonic oscillator (i.e. a spring). You're confusing the fact that spheres require quadratics but quadratics are not sufficient to admit a sphere.
For 1D harmonic oscillator, the sphere is 2D, and called a circle. It's rotating through time. 1D space + 1D time.
If changing definitions and adding dimensions are fair game, try it for a damped oscillator and tell me where whatever dimension sphere is in there.
If you understood what you were trying to push now you'd recognize that adding in your time dimension to force an argument involving a circle is just a matter of the sign in the 2nd order differential equation. This also falls apart if that sign changes and you get hyperbolic solutions. and I'm sure you'll just say that's just another kind of sphere with negative curvature
Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)
#228Earlier quoted context omitted.
For 1D harmonic oscillator, the sphere is 2D, and called a circle. It's rotating through time. 1D space + 1D time.
If changing definitions and adding dimensions are fair game, try it for a damped oscillator and tell me where whatever dimension sphere is in there. If you understood what you were trying to push now you'd recognize that adding in your time dimension to force an argument involving a circle is just a matter of the sign in the 2nd order differential equation. This also falls apart if that sign changes and you get hyper…
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Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)
#229Earlier quoted context omitted.
If you're talking about intuitions, you have no firsthand intuitions about lifting effort decreasing with distance to the Earth. We can intuit about constant gravity, and the math of constant gravity works fine for this description. And while the real situation at scale is more complicated, the math is going to come out to the same answer, albeit with extra terms muddying everything up. If someone says that something…
I’m not talking about intuitions; I’m talking against them. The intuition about carrying something 1st to 2nd then to 3rd floor is clearly wrong as evidenced by the example I gave; it is less wrong in smaller scales, but it still is wrong. If the floors were as high as the radius of the Earth, the first one would be three times as hard as the second one. The math doesn’t come out the same. It’s not at all linear, it’…
But the intuition based on the constant gravity assumption correctly illustrates the point, and adding in the more complicated picture of non-constant gravity creates new terms that exactly cancel out to give the same answer. Why would you talk against intuitions that work correctly?
And to be clear, building intuitions that fail at certain scales is still a useful and important thing to do. But you haven't shown a scale where these intuitions fail. That's not insightful, it's just throwing smoke bombs for no reason.
Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)
#230Earlier quoted context omitted.
If I follow you, that's not strange. That's exactly how Lagrangian mechanics are formulated (minimizing the action which has exactly the kinetic energy as a term to be minimized against a potential energy term) which rests on well-founded symmetry principles.
Action is linked to spatial symmetry too, and you can find the square there. Since space is isotropic, a Lagrangian can only depend on a speed vector through its norm. A Lagrangian must also be decomposable into independent orthogonal components, so you end up with an energy term that is shaped according to: f(√(a^2 + b^2)) = f(a) + f(b) And you end up with f being proportional to v squared. Note: the components do n…
That's a nice argument, though it doesn't hold relativistically