> 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.

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hunter2_ an hour ago | parent | next [-]

That's a good question, and I suppose the mgh formula isn't a suitable answer, so my answer would be something like: if you lift an object to some height, and then you repeat that action (lifting it from there to twice the height), you've done twice the work, and doing twice the work requires twice the caloric intake.

	▲	SilasX 8 minutes ago \| parent [-]
		Okay but that depends on the intuitions the question is trying to justify, which makes it circular. We also know, for example, that the body uses more than twice as much energy to do twice as much work (because of fatigue on the muscles or whatever the right term is here). In fact it takes positive energy just told a weight at a fixed height, doing zero mechanical work! So you’re actually appealing to even weaker intuition than the one the question is trying to ground!

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throwawaytea an hour ago | parent | prev | next [-]

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.

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nlawalker an hour ago | parent | next [-]

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.

	▲	throwawaytea 11 minutes ago \| parent \| next [-]
		That's clever, and I can't imagine or explain it as easily. Something to do with a reference point moving away from you so when solving for bringing it back to zero it's different than just adding the two energies back together. You have to add up the energy of catching them all up to the initial starting reference. I think also because distance is one unit, so moving reference pointe is easier. Moving reference points on distance over time already gets my spidey senses going that it's not something you should do without some real understanding.
	▲	hunter2_ 28 minutes ago \| parent \| prev \| next [-]
		When you're already going 10 mph and you're about to add another 10 mph, you can only "call that zero" (i.e., go from 0 mph to 10 mph again) if your point of reference (i.e., the ground) also begins moving with you at that point. Since the ground is stationary, you're definitely about to increase from 10 mph to 20 mph relative to the ground, not from 0 mph to 10 mph, and that's harder to do. But if you're on a treadmill that was stationary for the first change, and then suddenly starts moving at 10 mph right before the second change without affecting your speed relative to the ground, then you can "call that zero" and you'll be able to add another 10 mph (ending up at 10 mph relative to the treadmill and 20 mph relative to the ground) with the same ease as the first go.
	▲	cubic_earth 17 minutes ago \| parent \| prev [-]
		I suppose they are both "intuitive", but the example I gave was both intuitive and correct. Probably for anyone who has carried something or themselves up a hill, or climbed a set of stairs can relate to that from firsthand experience. I don't know what the kinetic energy corollary to that would be? "Stand still and I will throw a baseball at you going 15mph, and note how much it hurts. Now I will throw it at you going 30mph. See! It hurts 4x as much" :D

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pishpash 17 minutes ago | parent | prev [-]

Not really. Potential energy in a gravitational well obviously has absolute coordinates.

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gorgoiler 18 minutes ago | parent | prev | next [-]

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 :)

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raldi 19 minutes ago | parent | prev [-]

Because if the one falling 20ft lands on a seesaw, the other side of it will toss two balls each of the same mass 10ft up.

	▲	pishpash 12 minutes ago \| parent [-]
		Then 20ft should not be used in the explanation. They should just have one ball going at 2x speed hit the seesaw and have 4 of those balls go up at 1x speed.