# The final explanation to why kinetic energy is proportional to velocity squared

 PF Gold P: 2,022 I haven't the patience to read through all responses to this lengthy and ancient thread but surely the question has already been answered.If not here goes: From the conservation of energy the KE of an object will be equal to the work done in bringing that object from rest to a velocity v(or to bring it from a velocity v to rest) and is independent of the method by which that work is done. Work done=force*distance =mass* acceleration*distance(Mad)....(1) For uniform acceleration from rest to a velocity v in time t, a=v/t and d=(vt)/2 Substitute a and d into equation (1) and we get the well known non relativistic KE equation.
 P: 697 Without taking away from some of the good answers above, I'll add that Landau and Lifgarbagez answer this question right in the beginning of their famous Classical Mechanics books. This thread needed only one response. "Go look there and come back with any questions if it's not clear." Their explanation is clear and elegant. "Classics" are classics for the reason that everyone should read them.
 P: 777 It's because if you want to make something go twice as fast, you have to either push it 4x as hard, 4x as far...or some combination of these. Whichever path you take, you'll get 4x as tired. There's no other way around it. Since energy is force x distance, it stands to reason that as speed increases then so will the distance. Simple math show exactly how much the distance increases.
PF Gold
P: 2,022
 Quote by dnquark This thread keeps coming back because nobody gave an entirely satisfactory explanation that motivates v^2 proportionality. (I can't give one either, I came here in search of one). What puzzles me is all the explanations that involve work-energy theorem. They make sense mathematically, but they require you to define an auxiliary quantity called "work" with dW = F dx. How is that definition more intuitive than defining energy as ~ m v^2?.. I think the original question should be rephrased as "what is the minimum set of assumptions one needs to make in order to identify 1/2 m v^2 as a conserved quantity?" One candidate answer is assuming the principle of least action, from which energy conservation (and KE expression) can be derived using Noether's theorem. However, that's probably not the only possible answer. For instance, Leibnitz and du Chatelet probably had other motivations when they posited v^2 dependency of energy.
I have just scanned through other posts on this thread and your post,in particular,has caused me to re-evaluate my post 55.I agree that if energy is defined with respect to work then it is no more intuitive than defining it with respect to (mv^2)/2 or with respect to anything else that is relevant such as gravitational potential energy.It's the chicken and egg,circular argument syndrome.Some people may have an intuitive feeling for energy but I think that it is partly(perhaps fully)because those people have gained some familiarity with the subject.We observe energy but I don't think we really understand it.Why then is KE proportional to v squared?I now think that a possible best answer is quite simply that "it is because observations and experiments show it to be so".
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P: 15,167
 Quote by dnquark This thread keeps coming back because nobody gave an entirely satisfactory explanation that motivates v^2 proportionality.
Read the thread. It follows directly from the definition of work. The derivation has been given many times; post #25 gives a very nice and succinct derivation.

This begs the question, why define work that way? As is the case with many of the widely used concepts in physics (why is force defined to be mass times acceleration?), the answer is that we use this definition because it is so very useful and so very simple. We physicists are very much enamored with utility and simplicity, particularly when expressed mathematically, and particularly so when it describes some aspect of reality.

The concept of energy was an important one even in basic Newtonian mechanics. The concept took on an even bigger role with the developments of thermodynamics and in the development of the Lagrangian/Hamiltonian formulations of Newtonian mechanics. The concept of energy plays a predominant role in quantum mechanics. Noether's theorems pretty much iced the cake.

The concept does require a bit of modification with regard to special relativity. With these mods it remains a very important, if not the central, concept of physics.

Bottom line: We physicists use the concept of energy because it works so very nicely for us (pun intended).
PF Gold
P: 2,022
 Quote by D H Read the thread. It follows directly from the definition of work. The derivation has been given many times; post #25 gives a very nice and succinct derivation. This begs the question, why define work that way? As is the case with many of the widely used concepts in physics (why is force defined to be mass times acceleration?), the answer is that we use this definition because it is so very useful and so very simple. We physicists are very much enamored with utility and simplicity, particularly when expressed mathematically, and particularly so when it describes some aspect of reality. Bottom line: We physicists use the concept of energy because it works so very nicely for us (pun intended).
I have to agree.Experiments reveal that there is a quantity which we call energy which can appear in several different forms and which is conserved and which has other observeable properties.It may be possible to define energy in different ways but given the choice we should go for the definition which is the most elegant,most useful and most simple to use and I think the definition of "energy"in terms of "work" meets these criteria.
P: 66
 Quote by DaleSpam Because the units wouldn't work out otherwise. When I was taking freshman physics that is what my professor pounded into our heads the first week or two: "always check the units". Kinetic energy couldn't possibly be anything other than kmv² where k is some unitless constant.
that was the point of my post - dimensional analysis

if the velocity was raised to any other power except for 2 the formula collapses into the abyss of invalidness - where all the dead ends are

All formulae must pass the dimensional consistency test

Even the weird ones from quantum mechanics
 P: 2 I'm no physicist per se, but I find this subject very interesting. I am too one of those who are inclined to revise and better understand the basics first, then go into more complicated matters. All people here seem to be quite advanced in the mathematics of physics which I kept avoiding all the time :) When reading this thread, an image kept popping up in my head. The image of getting away from a balance point. Maybe we require more force to accelerate an object from 50 mph to 100 mph (than from 0 to 50) because it's much further from the balance point, and another natural force drags the object backwards (where backwards is opposite to speeding up). Like gravity when we go upwards. Maybe the object needs more force because the balancing force is growing rapidly. Maybe we need to apply then more force, to cope with the backwards balancing force. An interesting logical result (for me at least), is that the object and everything else must be connected somehow. Because forcing a separated object doesn't affect anything else but itself => there would be no need for speed squared in the formula. That's all I have to say for now
 P: 2 We all know from reality that a car has much more than a double damage when it crashes at 100 km/h instead of 50 Km/h. So speed must be considered more than first power. Let's choose second power: it works! So why don't accept it?
 P: 2 :) what you say it's like: if you found cause(n-1), why bother to know cause(n-2) ? why did you bother to find cause(n-1) in the first place?
 P: 2 Just wanted to say it cannot be first power because not enough. I bother about the topic, I'm still thinking about it and haven't accepted it, to be honest. Sometimes when mathematics goes too far I loose the touch of reality. It's my limit. Alberto
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P: 11,778
 Quote by DaleSpam Hehe, I give this thread the zombie "night of the living thread" award. It first died in 2005, came back to life for a day in 2008 and promptly died again, and then came back to life again in 2010 where it has been terrorizing the villagers for a couple of weeks now!
It's time to use the wooden stake. Whack! Whack! Whack!

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