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I tried to understand why kinetic energy is proportional to the square of velocity. In this endeavor I stumbled upon a book "Emilie du Chatelet: Daring Genius of the Enlightenment" (ISBN 978-0-14-311268-6), where she explains it in one section. The key point I took away is "...a moving body accumulated force, and thus the formula describing this movement must include squaring of the speed." I understand that the statement faster moving bodies "accumulate" even more force implies squaring of the speed, but I don't understand why the initial statement is true. Also, I know that a body with twice the speed will penetrate four times deeper upon collision, but that is a demonstration, not an explanation.

Henry05
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    I don't think any modern physicist would talk about "accumulated forces". 18th century physicists, however genius they might have been, worked in very different contexts than modern physics, and there is considerable expertise necessary to translate their thoughts into a modern world view. Do you want someone to explain 1. what this particular person might have meant by "accumulating force" or 2. how modern physics thinks about kinetic energy? These are different questions with potentially very different answers. – ACuriousMind Nov 19 '22 at 15:31
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    For the latter question, this would be a duplicate of https://physics.stackexchange.com/q/535/50583. For the former question, it might be more appropriate for [hsm.SE], but not off-topic here. – ACuriousMind Nov 19 '22 at 15:32
  • Thank you for your answer. I've already read that article, but both explanations are not satisfactory for me (assuming I understood them correctly) - first one involves demonstration, which proves it, but it doesn't help me understand it (unless I missed something). The second one is also a proof involving torque, which I also have a problem with understanding, because the only proof involves work, but I don't really understand work and kinetic energy in the first place. What I read in the book was hitting the nail exactly on what I didn't understand. – Henry05 Nov 19 '22 at 15:52

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I think Emilie meant this (based on a documentary) :

The maximum height reached by the ball, going upward against Earth's gravity with speed $v$ is given by (assuming it starts at height 0):

$$s=\frac{v^2}{2g}$$

So, if a ball has twice the speed of another ball, it reaches four times the height that the other ball reaches.

Today, we understand this in terms of the work-energy theorem. Gravity has to do work, on the twice-as-fast ball, for four times as long a distance, to convert all of the ball's kinetic energy into potential energy.

Emilie showed the mass times square of the speed of an object was a useful physical quantity, as a "measure of how much motion the object carries", where the "measure of motion" means "how much work it takes to stop the object"

Today, we don't call this quantity "accumulated force". We call it " Kinetic energy".

Ryder Rude
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  • Thank you for your answer. Your answer contains a demonstration that also relies on "a constant force slowing down an object with twice the speed will stop it at four times the distance", which still doesn't help me understand. But I recognize the formula you used (one of the kinematic equations) and I think that is the reason I don't understand kinetic energy - because I don't understand that kinematic formula in the first place. – Henry05 Nov 19 '22 at 16:07
  • @Henry05 I think what you really need is an intuitive derivation of either the work-energy theorem, or this kinematic formula. Have you seen the derivations? – Ryder Rude Nov 19 '22 at 16:09
  • I have seen them, but I only understand the derivation that gets me s=1/2 gt^2. – Henry05 Nov 19 '22 at 16:24
  • @Henry05 The only philosophical argument that I can think of, for why work is an important quantity, is that our universe has conservative forces. This is an experimental discovery which motivates "work" as a useful physics quantity. From there on, the equality of work and kinetic energy is derived using purely mathematical manipulation. The square on $v$ comes comes from the fact that, as part of the derivation, we want to calculate the are under the straight line $f(v) =v$. The area is $\frac{1}{2} v^2$ using the traingle area formula – Ryder Rude Nov 19 '22 at 16:56
  • @Henry05 First, you should read about conservative forces. Their existence inspires us to derive an expression for work. Using some manipulation, we end up with $vdv=adx$. See my answer here. After "integrating both sides", which means "taking the area under a curve", we derive the kinetic energy formula. In the end, it's just a useful result for calculation. Don't overthink it for a deep philosophical meaning – Ryder Rude Nov 19 '22 at 17:01