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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)

#191

Earlier quoted context omitted.

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.

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

#192

Earlier quoted context omitted.

But what if the cars are spherical cows?

Cows can't roll that fast.

Cows don't roll - they are frictionless so just glide over the surface smoothly.

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

#193

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

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).

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

#194
Brilliant answer. It is not intuitive that while it takes the same energy to heat an object from 0 C to 1 C as it does from 1 C to 2 C, it takes 1/3 of the energy to speed up an object from 0 kph to 1 kph than it does from 1 kph to 2 kph.

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

#195

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.

"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.

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

#196

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.

> 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..,

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

#197

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

10ft vs 20ft is factor of 2 difference in height, 1ft vs 10ft is factor of 10.

Also how it translates to difference in damage depends on how elastic is the collision. If the balls are made of rubber, things get much less dramatic, and energy exchange ratios more obvious.

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

#198

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

Dropping something from 1 foot has 1/10th the kinetic energy compared to the same thing dropped from 10 feet. Damage is a very poor proxy for energy because there are all kinds of variables and thresholds and structural consideration in determining damage.

A cup might not break at all in a 1 foot fall and might shatter when dropped from 10. The outcome is binary, which isn't useful as a scale of input energy.

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

#199
Why would you expect the concept of kinetic energy to be intuitive? Where would an intuition for a mathematical invention come from? It is a name for a useful mathematical relationship. Are negative or imaginary numbers intuitive, even when you understand how they function?
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