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

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

#81

Post by gorgoiler »

nlawalker wrote: post #61
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. 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.

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 :)

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

#82

Post by cubic_earth »

nlawalker wrote: post #72
throwawaytea wrote: post #69

Earlier 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.

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

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

#83

Post by pishpash »

throwawaytea wrote: post #69
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.

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.

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

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

#84

Post by acchow »

c1ccccc1 wrote: post #56

A stationary but hot object has kinetic energy due the the motion of the individual atoms that make it up, even though its overall momentum is 0. I.e. ∑ⱼ mⱼ v⃗ⱼ = 0⃗ where the mⱼ are the masses of the parts of the object and the v⃗ⱼ are the velocities of those parts. If the object initially has 0 velocity, its kinetic energy is: T = ½∑ⱼ mⱼ v⃗ⱼ² Now we give the object a kick (or just switch reference frames) to change…

Looks like in your 2nd equation you've already assumed kinetic energy is quadratic with speed

T = ½∑ⱼ mⱼ v⃗ⱼ²

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

#85

Post by pishpash »

raldi wrote: post #79
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.

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.

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

#87

Post by throwawaytea »

nlawalker wrote: post #72
throwawaytea wrote: post #69

Earlier 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.

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.

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

#88

Post by gorfian_robot »

AngryData wrote: post #28

This is also why splitting wood with a maul is way more work than using an axe. You can swing an axe at incredibly speeds which gives incredibly transfers of energy, but a maul is going to always have "meh" levels of speed because it is too much mass to accelerate over such a short distance as a swing. Also why you don't see framers using 3 lb hammers. You can put in more effort and get your lighter hammer swing to t…

have you ever had to split wood?

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

#89

Post by SilasX »

hunter2_ wrote: post #63
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.

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!

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

#90

Post by zkmon »

No amount of scientific explanation can exhaustively explain a phenomenon. Feynman puts this nicely with the story of "Why did aunt slipped and fell down" in his talk about magnetism.

For instance we know that the life forms grow via cell division, but no text can address the question of "why". They can only talk about "how".

Infact, science quest is not really about answering "why" all the way down the causal chain. It is about learning how the qualities of things are related and a bit of shallow causal chain inspection.

The causal chain, by nature, does not allow full inspection. It's dependency on temporal constructs means it breaks down where time breaks down. Infact causality might might break down at macro levels as well, leading to loops with no end or beginning (kind of chicken and egg problem).

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