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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.

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!

All intuitions are wrong, but some are usefull. You have to do the experiment, discover the formula, and then adapt your intuitions accordingly.

What point that I made are you responding to? I was disputing someone’s appeal to a specific intuition for being an unhelpful one to use here. So I obviously get the concept of some intuitions being useful. Did you see the comment being responded to?

At some point you just have to shut up and calculate.

I think if you define energy as force X distance then integration alone will give you the squared term.

How I got banned from some reddit channel. Flip this around ask if a ball were fired out of a gun up into the air what height would it reach? A ball twice as fast goes up 4 times as high. If energy is force times distance it had 4 times the energy.

> In fact it takes positive energy just told a weight at a fixed height, doing zero mechanical work!

Stacking a weight on top of a table holds it at a fixed height and requires zero mechanical work.

The failure in intuition here relates to physiology and the mechanism by which muscles work, not physics. Myosin and actin are constantly cycling through bonding and release during muscle contraction, as this is how the shortening action actually occurs. In fact, muscle contraction is particularly unintuitive because people frequently consider ATP the "energy currency", yet the ATP-consuming steps are actually the release/relaxation and preparation for binding, not the pulling action. This is also why the phenomena of rigor mortis upon death occurs.

I get that involving a human body complicates the analysis. That was the point: that you can’t appeal to it as a simple example to ground the intuition in other case.

Also:

>Stacking a weight on top of a table holds it at a fixed height and requires zero mechanical work.

I was referring to a human holding it. What would have been a better way to keep you from missing that?

> I get that involving a human body complicates the analysis. That was the point: that you can’t appeal to it as a simple example to ground the intuition in other case.

Yeah, fair enough. It's unfortunate that the comment you were responding to involved "caloric intake" suggestive of a biological system when it could just as easily have involved a mechanical pulley. Their intuition would have been stronger phrased as: Within a gravitational field that is approximated as a uniform force field, the amount of energy/work to raise a weight by a fixed distance is independent of the initial position of that weight on a (massless) rope attached to a (frictionless) pulley.

> I was referring to a human holding it. What would have been a better way to keep you from missing that?

Since you are asking, for me, it would helped to have the following inserted text: "In fact it takes positive energy [for biological muscle] just told a weight at a fixed height, doing zero mechanical work!" so that the statement stood on it's own, or else including within your post a more definitive statement to the effect of "intuitions involving the human body complicates the analysis", as you did in this reply. If "that was the point", go ahead and state it.

To be fair, I often have the same problem. It's easy for me to write a lot of text that goes around a statement without actually getting to it.

> 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.

You’re introducing two new intuitions, and it’s not intuitively obvious how they are related to each other. Why would work correlate 100% with caloric intake, and caloric intake 100% with kinetic energy?

Certainly, ‘work’ is highly counterintuitive. If I move a concrete block over loose sand on a beach, I’m doing zero work, in the physics definition, so moving it over a kilometer should be as easy as moving it for a millimeter.

Even ignoring the difference between caloric intake and caloric expenditure, it also isn’t intuitive to me that caloric expenditure is independent of the speed at which one lifts an object.

In the end, the answer is “because the math works out that way, and kinetic energy is a useful concept”

Friction. Work isn't just about height.

Holding that block stationary at arms length then. 0 work.

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.

"Work" is the weird thing in physics, I'd say is about as opposite to intuitive as you can get when introducing a concept. It's only intuitive when considering lifting an object - say, a bag of groceries. Heavier the bag, higher the lift -> more work. But then you carry that heavy bag a couple kilometers, arrive at home exhausted, only to be told by the physics teacher that you did exactly 0 work. Or in fact negative work, if upon coming home, you put the bag down.

I understand the concept myself somewhat intuitively now, but that intuition is not connected to everyday experience - it's just familiarity with a detached concept of physics!work that just is what it is, but is consistent in being that.

Well, a good physics teacher might note the difference between external work on the gravitational field, versus the internal thermodynamic work of the human body. Which could lead to a quick discussion of how muscles are a dynamic system depending on constant activation of actin & myosin (and therefore consumption of ATP) as opposed to a static elastic system like a piece of metal.

Very intuitive analogy is that running an engine in neutral burns fuel, but doesn't do work to move the car forward.

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.

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.

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

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.

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.

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

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.

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.

That ends up begging the question, because the next step is "how high do you have to drop it from so that it's travelling twice as fast?" and you're immediately going round in circles.

Nothing of the sort. The seesaw can be in space far from gravitational influences. Potential energy is extraneous in this explanation.

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.

> Going from A to B then B to C is the same as going from A to C.

Not really, no. Not all forces are conservative.

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 intuition for kinetics. Increasing walking/running from 0 to 5 km/h doesn’t feel the same as than moving from 5 to 10, which does not feel the same as moving from 10 to 15. I don’t think we have an experience of linear relationship between running speed and effort, or other types of speed/energy types of relationships.

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

I guess Galileo came up with it first:

https://en.wikipedia.org/wiki/Galilean_invariance

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.

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

and also pushing that reference frame down when moving up

That's a good point. For a stationary observer sitting on the ground it would roughly seem that, in the airplane case, you increase your speed from 100s mph to `100s + ε mph`, while for the home case from 0 mph to ε mph. So that seems like a counterexample to what I described as common kinetic experience.

I think the issue here is that, in order to move, you apply force to the floor of the airplane. Because the airplane has huge mass and your mass and relative speed are minuscule, there is (probably) no perceivable effect on the airplane's motion. So you increase your kinetic energy by the same amount in both cases while expending the same amount of (chemical) energy, but in the airplane case, the kinetic energy of the airplane (just the airplane, without you) decreases (by a miniscule amount compared to its actual kinetic energy, but still).

Scale up the numbers in you example: The effort to move a piece of furniture from 10,000th to 20,000th floor is NOT the same as the effort to move it from the 20,000th to the 3rd. The reduced gravity will help you.

On earth, it just about is... you haven't scaled up enough. Low earth orbit doesn't have much less gravity, it's just that there's no air resistance so you can move fast enough sideways so that you don't run into the earth. Hence orbit and not just floating.

But more to the point the kinetic energy here is being turned into gravitational potential energy. If you move to a place with a weaker gradient in gravitational potential of course the same amount of kinetic energy moves you farther up.

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 true can be illustrated intuitively with a thought experiment, "sure, but what if we take that to a scale where our intuitions fail" is a sort of odd place to take the discussion unless you're genuinely curious how the math is going to shake out.

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’s the inverse square; that’s much more than just _extra terms muddying things_.

Calling this relation linear by just looking at the intuitions of tiny humans is akin to hyper-zooming an exponential graph and calling it linear. It is “approximately true” locally, but hey, the same is also true for velocity vs kinetic energy!

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 object moving 0.0001 MPH.

So yeah, I am zooming in on an exponential and calling it linear, but that doesn't work for speed. And doubling of velocity gives 4x the energy regardless of how tiny the step.

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.

What intuitive understanding do you have of moving furniture up 10,000 floors? None.

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.

> What makes this intuitive?

20 million years of evolution hard-wiring it into our primate brains on a genetic level, from every thrown rock and fall from a tree. That's what made it intuitive. But not everyone gets the same batch of genes, I guess..,