Why does kinetic energy increase quadratically, not linearly, with speed? (2011) • Hacker News Board

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#161

Post by justsid » Sat, Jun 27, 2026, 2:46 PM UTC

trinari wrote: post #158

freehorse wrote: post #117

Earlier quoted context omitted.

The effort to move a piece of furniture from 1st to 2nd floor is the same as the effort to move it from the 2nd to the 3rd. We have good intuition for this by our experience, which derives a linear relationship. The effort to move a piece of furniture up two floors is double the effort of moving it up one floor (ie you have to put the same effort twice, assuming enough rest). I would not say we have the same intuitio…

Can someone help me understand the following? Getting up from a seat als walking a couple steps feels that same at home and in a flying airplane (or does it?). But the base speed is 0 in the former and several hundred mph in the latter case

When you get up from a seat and walk a few steps you are already doing that on something that is hurtling down space. We don’t notice that our planet moves a lot, because we can’t really see the movement in our reference frame. If you were on a plane without any windows, no turbulence and no sound cues from the engine, you wouldn’t know when getting up from your plane seat that you are in a moving object either.

Acceleration is a real force that we can feel. But once moving at a constant speed, physics dictates that it’s all the same. That’s also why you can throw a tennis ball up on a plane and not have it fly backwards immediately smacking into the person behind you.

In the reference frame of you and the aircraft, you are not moving at all and neither is the plane. In the reference frame of the ground you and the plane are moving.

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#162

Post by Xmd5a » Sat, Jun 27, 2026, 2:52 PM UTC

[flagged]

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#163

Post by GistNoesis » Sat, Jun 27, 2026, 3:01 PM UTC

terminalbraid wrote: post #147

GistNoesis wrote: post #114

Every time in physics you see quadratic, you should think sphere. There is some rotation invariance hidden in the velocity physics because you can rotate the velocity vector of an object without having to spend energy (The force you need to apply is perpendicular to the velocity so does no work). The typical example is you have a ball fall 1m vertically, then have a 90° bend which convert the vertical velocity into h…

> 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.

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#164

Post by PunchyHamster » Sat, Jun 27, 2026, 3:12 PM UTC

cubic_earth wrote: post #55

It'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…

> We know intuitively that a ball atop a 20ft ladder has twice the potential energy of a ball atop a 10ft ladder.

...no ? dropping something 10 times from 1ft is nowhere near energetic/damaging as once from 10tf

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#165

Post by PunchyHamster » Sat, Jun 27, 2026, 3:14 PM UTC

trinari wrote: post #158

freehorse wrote: post #117

Earlier quoted context omitted.

The effort to move a piece of furniture from 1st to 2nd floor is the same as the effort to move it from the 2nd to the 3rd. We have good intuition for this by our experience, which derives a linear relationship. The effort to move a piece of furniture up two floors is double the effort of moving it up one floor (ie you have to put the same effort twice, assuming enough rest). I would not say we have the same intuitio…

Can someone help me understand the following? Getting up from a seat als walking a couple steps feels that same at home and in a flying airplane (or does it?). But the base speed is 0 in the former and several hundred mph in the latter case

you are still moving against reference frame (floor) that is at speed 0.

and also pushing that reference frame down when moving up

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#166

Post by p-e-w » Sat, Jun 27, 2026, 3:15 PM UTC

mike_hock wrote: post #142

p-e-w wrote: post #104

Earlier quoted context omitted.

It wasn’t “prescient”, StackExchange sites have always been among the most hostile communities on the Internet. Today they are additionally weighed down by increasingly erratic management decisions desperately trying to extract as much monetary value as possible before AI completely obsoletes SE, but the amount of aggression and hostility on the network was unbearable from the start. I remember dozens of occasions wh…

Farmers are desperately trying to milk (hah) as much monetary value as possible from their cows before packaged milk obsolesces them completely!

The analogy doesn’t quite work because it’s not clear whether LLMs actually need additional mediocre-quality, human-written explanations in order to improve, or whether better training pipelines and ingesting documentation might be enough.

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#167

Post by PunchyHamster » Sat, Jun 27, 2026, 3:18 PM UTC

WithinReason wrote: post #116

gorgoiler wrote: post #81

Earlier quoted context omitted.

Because physical movement is intuitively transitive. Going from A to B then B to C is the same as going from A to C. The journey from Y to Z might feel more tiring than the journey from A to B, but only if you do them all in one day :)

So why isn't increasing your velocity from A to B then B to C the same as A to C? Isn't that intuitively transitive too?

it is if your reference point is A in both cases.

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#168

Post by Xmd5a » Sat, Jun 27, 2026, 3:31 PM UTC

terminalbraid wrote: post #149

sixo wrote: post #70

Kinetic energy is, strangely, quite a bit like a least squares cost function in an optimization problem. The "dt"s in "dx/dt" hardly matter; it basically represents "dx^2" between the current state and the next.

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 not need to be independent and orthogonal for this to hold.

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#169

Post by netbioserror » Sat, Jun 27, 2026, 3:32 PM UTC

I recently learned, through visual intuition, how the relative perception of time between two subjects changes as relative speed between them changes. It's because they are observing each other from an "angle" in the time dimension. And in that time dimension, angles do not trace circles, they trace paraboloids.

If I'm remembering correctly, this is also why the energy required to "reach" the speed of light for subjects with mass parabolically goes to infinity. I'm also guessing it can directly trace a proof down to why kinetic energy increases quadratically.

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#170

Post by cubic_earth » Sat, Jun 27, 2026, 3:32 PM UTC

njstraub608 wrote: post #122

nlawalker wrote: post #61

Earlier quoted context omitted.

> We know intuitively that a ball atop a 20ft ladder has twice the potential energy of a ball atop a 10ft ladder. What makes this intuitive? The foundation of the asker’s question is that it seems intuitive that kinetic energy would increase linearly with speed, but that turns out to be wrong.

Feels like what OP meant to say is, “you could rightly assume that a ball…” instead. Seems like a fair starting point if you’re just doubling things because if the height difference. I really liked cubic’s explanation overall.

That indeed would have been better. Much too late for that edit now. But the subsequent debate over the intuitive claim is fun.