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.
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2$\begingroup$ 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. $\endgroup$– ACuriousMind ♦Commented Nov 19, 2022 at 15:31
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2$\begingroup$ For the latter question, this would be a duplicate of physics.stackexchange.com/q/535/50583. For the former question, it might be more appropriate for History of Science and Mathematics, but not off-topic here. $\endgroup$– ACuriousMind ♦Commented Nov 19, 2022 at 15:32
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$\begingroup$ 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. $\endgroup$– Henry05Commented Nov 19, 2022 at 15:52
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2 Answers
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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".
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$\begingroup$ 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. $\endgroup$– Henry05Commented Nov 19, 2022 at 16:07
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$\begingroup$ @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? $\endgroup$ Commented Nov 19, 2022 at 16:09
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$\begingroup$ I have seen them, but I only understand the derivation that gets me s=1/2 gt^2. $\endgroup$– Henry05Commented Nov 19, 2022 at 16:24
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$\begingroup$ @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 $\endgroup$ Commented Nov 19, 2022 at 16:56
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1$\begingroup$ @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 $\endgroup$ Commented Nov 19, 2022 at 17:01
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I intuitively expected the kinetic energy to be the same as momentum. The reason why it cannot be the same lies in the fact that kinetic energy (gained by doing work) is the total force accumulated over some distance, since $$E_k = W = \int F\mathrm d l$$ while momentum is the accumulation of force over time, since from the Newton's second law we get $$\frac{\mathrm d P}{\mathrm d t} = F\to P = \int F\mathrm dt$$ If we set a body in motion from standstill with force $F$ so it gains the speed $v$, it will accumulate a certain amount of force over time and distance $l$. If we use the same force again to double the speed, the force accumulated over this time interval does not depend on $v$. But, since the object was already moving, the total distance travelled in the second "phase" must be the distance the object would travel if it was unaccelerated with speed $v$, plus, the distance the object would travel if it was accelerated by the force $F$ but did not have the initial speed $v$. Thus, the total distance traveled must be necessarily longer and it would also be convenient to introduce a quantity that reflects this increased growth of accumulation of force with respect to distance caused by a dependence on instantaneous speed.
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$\begingroup$ It is correct. And again, the big reason we introduce the quantities "accumulation of force over time/distance" is the conservation of momentum/energy respectively. Both are experimental facts that follow from Newton's third law and the existence of conservative forces respectively. $\endgroup$ Commented Jun 12 at 16:15