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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)
#112this is implicitly an is-ought question and it is pedagogically necessary to give an unsatisfying answer. you can watch the interview with feynman answering "why do two objects attract" for the best version. The top answer selects an arbitrary resolution in the hierarchy of the unlimited series of 'whats and whys' but the true answer to any why question in physics is "because it is".
Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)
#113[dead]
Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)
#114Every 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 horizontal velocity and no vertical velocity, then the ball fall again 1m vertically and have its vertical velocity increased by the same amount as for the first meter. You can then add a 45° degree bend ramp to redirect the ball so that it only has horizontal velocity, and have the ball fall again. For the third bend ramp the incoming velocity will have 2 units horizontal, and 1 unit vertical (I'll let you compute the appropriate angle). A fourth ramp would be 3 units horizontal and 1 unit vertical.
Because we can do this adding velocity in a perpendicular way trick we must then use Pythagoras.
Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)
#115walking into a wall slowly doesn't hurt much, but you really don't have to speed up a lot for it to a hurt a whole lot more.
Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)
#116Earlier 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 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?
Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)
#117It'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.
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.
Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)
#118Fun little anecdote: A blue care is travelling along at 70 units, and a red car (exact same make and model) is catching up to it going 100. When they're both right beside each other a bend in the road reveals an obstacle blocking both lanes, so both cars brake at the same intensity and deceleration. The blue care stops right before the obstacle. Since the red car was going at a faster speed, and braked at the same ra…
There's a great Australian traffic safety ad that makes this same point: https://www.youtube.com/watch?v=7x7c0qNGbv0
Nice bit of camera trickery. He says "both drivers react and a moment later they break", but the cars are still side by side. It (apparently) takes drivers 1.5 seconds to respond, the 5 km/h speed difference cuts the distance by 2 meter. Which apparently is a big deal. Rough estimate breaking distance:
5 km/h = 0.13 meter
30 km/h = 4.5 meter
60 km/h = 14 to 18 meter
65 km/h = 21 to 24 meter
You need 150% the distance at 65 vs 60.
Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)
#119A similar question I asked a few years ago: https://physics.stackexchange.com/questions/740056/how-much-...
Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)
#120LOL Kinetic energy increase quadratically for sub-relativistic speeds only.
Kinetic energy
E = (m * v^2)/2 + (3*m * v^4 )/8*c^4 + (5*m * v^6)/16*c^6 …
and so on, so kinetic energy increases infinitely faster than speed, thus it impossible to reach c, because it requires infinite amount of kinetic energy.
Why? Because of rules of wave propagation.