# Why does kinetic energy increase quadratically, not linearly, with speed?

As Wikipedia says:

[...] the kinetic energy of a non-rotating object of mass $m$ traveling at a speed $v$ is $mv^2/2$.

Why does this not increase linearly with speed? Why does it take so much more energy to go from 1 m/s to 2 m/s than it does to go from 0 m/s to 1 m/s?

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It doesn't increase exponentially (a^x); it increases quadratically (x^2). Can you correct the title? – nibot Nov 11 '10 at 0:09

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What I found most counter-intuitive about this is that kinetic energy is a coordinate-dependent quantity. In a different inertial frame, the object will have a different kinetic energy. This contradicts the intuitive notion that kinetic energy should be independent of the coordinate system. – nibot Nov 11 '10 at 0:12
Did you mean quadratically ($v^2$)? Exponentially means something different ($~\exp(v)$). I believe there is derivation in every standard high school level textbook ($dE = F dx = m a dx = m \frac{dv}{dt} dx = m \frac{dx}{dt} dv = m v dv$). – Piotr Migdal Nov 11 '10 at 0:14
I took care of fixing the title for you, @Generic Error (hope you don't mind). – David Zaslavsky♦ Nov 11 '10 at 0:36
I'm tempted to answer...because it just is? Really, it's the way the universe works. – Noldorin Nov 11 '10 at 0:47
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## 13 Answers

The question is especially relevant from a didactical point of view because one has to learn to distingish between energy (work) and momentum (quantity of motion).

The kinematic property that is proportional to $v$ is nowadays called momentum, it is the "quantity of motion" residing in a moving object, it's definition is $p:= mv$.

The change of momentum is proportional to the impulse: impulse is the product of a force $F$ and the timespan $\Delta t$ it is applied. This relation is also known as the second law of Newton: $F \Delta t = \Delta p$ or $F dt = dp$. When one substitutes $mv$ for $p$ one gets it's more common form: $F= m \frac{\Delta v}{\Delta t} = ma$.

Now for an intuitive explanation that an object with double velocity has four times as much kinetic energy.
Say A has velocity $v$ and B is an identical object with velocity $2v$.
B has a double quantity of motion (momentum) - that's were your intuition is correct!
Now we apply a constant force $F$ to slow both objects down to standstill. From $F \Delta t = \Delta p$ it follows that the time $\Delta t$ needed for B to slow down is twice as much (we apply the same force to A and B). Therefore the braking distance of B will be a factor of 4 bigger then the braking distance of A (its starting velocity, and therefore also its mean velocity, being twice as much, and its time $\Delta t$ being twice as much, so the distance, $s = \bar{v}\Delta t$, increases 2x2=4 times).
The work $W$ needed to slow down A and B is calculated as the product of the force and the braking distance $W=Fs$, so this is also four times as much. The kinetic energy is defined as this amount of work, so there we are.

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Thank you for a nice clear explanation. – Generic Error Nov 12 '10 at 6:53
for years I've had this question about what momentum is that makes it so special in the second law. Thank you for explaining it so simply. – Preet Sangha Jun 21 '11 at 1:42

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This answer is vacuous, work is force times distance is synonymous with, and equally mysterious as, energy is half momentum times velocity, but at least it isn't wrong. – Ron Maimon Oct 28 '11 at 22:36
Agree with the comment above. – Diego Jan 26 '12 at 8:14

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@maimon: your own answer is indeed very interesting; the answer above has its own merit. – Gerard May 4 '12 at 12:42

The previous answers all restate the problem as "Work is force dot/times distance". But this is not really satisfying, because you could then ask "Why is work force dot distance?" and the mystery is the same.

The only way to answer questions like this is to rely on symmetry principles, since these are more fundamental than the laws of motion. Using Galilean invariance, the symmetry that says that the laws of physics look the same to you on a moving train, you can explain why energy must be proportional to the mass times the velocity squared.

First, you need to define kinetic energy. I will define it as follows: the kinetic energy E(m,v) of a ball of clay of mass m moving with velocity v is the amount of calories of heat that it makes when it smacks into a wall. This definition does not make reference to any mechanical quantity, and it can be determined using thermometers. I will show that, assuming Galilean invariance, E(v) must be the square of the velocity.

E(m,v), if it is invariant, must be proportional to the mass, because you can smack two clay balls side by side and get twice the heating, so

$$E(m,v) = m E(v)$$

Further, if you smack two identical clay balls of mass m moving with velocity v head-on into each other, both balls stop, by symmetry. The result is that each acts as a wall for the other, and you must get an amount of heating equal to 2m E(v).

But now look at this in a train which is moving along with one of the balls before the collision. In this frame of reference, the first ball starts out stopped, the second ball hits it at 2v, and the two-ball stuck system ends up moving with velocity v.

The kinetic energy of the first ball is mE(2v) at the start, and after the collision, you have 2mE(v) kinetic energy. But the heating is the same, so

$$mE(2v) = 2mE(v) + 2mE(v)$$

$$E(2v) = 4 E(v)$$

which implies that E is quadratic.

### Noncircular force-times-distance

Here is the noncircular version of the force-times-distance argument that everyone seems to love so much, but is never done correctly. In order to argue that energy is quadratic in velocity, it is enough to establish two things:

• Potential energy on the Earth's surface is linear in height
• Objects falling on the Earth's surface have constant acceleration

The result then follows.

That the energy in a constant gravitational field is proportional to the height is established by statics. If you believe the law of the lever, an object will be in equilibrium with another object on a lever when the distances are inversely proportional to the masses (there are simple geometric demonstrations of this that require nothing more than the fact that equal mass objects balance at equal center-of-mass distances). Then if you tilt the lever a little bit, the mass-times-height gained by 1 is equal to the mass-times-height gained by the other. This allows you to lift objects and lower them with very little effort, so long as the mass-times-height added over all the objects is constant before and after.This is Archimedes' principle.

Another way of saying the same thing uses an elevator, consisting of two platforms connected by a chain through a pulley, so that when one goes up, the other goes down. You can lift an object up, if you lower an equal amount of mass down the same amount. You can lift two objects a certain distance in two steps, if you drop an object twice as far.

This establishes that for all reversible motions of the elevator, the ones that do not require you to do any work (in both the colloquial sense and the physics sense--- the two notions coincide here), the mass-times-height summed over all the objects is conserved. The "energy" can now be defined as that quantity of motion which is conserved when these objects are allowed to move with a non-infinitesimal velocity. This is Feynman's version of Archimedes.

So the mass-times-height is a measure of the effort required to lift something, and it is a conserved quantity in statics. This quantity should be conserved even if there is dynamics in intermediate stages. By this I mean that if you let two weights drop while suspended on a string, let them do an elastic collision, and catch the two objects when they stop moving again, you did no work. The objects should then go up to the same total mass-times-height.

This is the original demonstration of the laws of elastic collisions by Christian Huygens, who argued that if you drop two masses on pendulums, and let them collide, their center of mass has to go up to the same height, if you catch the balls at their maximum point. From this, Huygens generalized the law of conservation of potential energy implicit in Archimedes to derive the law of conservation of square-velocity in elastic collisions. His principle that the center of mass cannot be raised by dynamic collisions is the first statement of conservation of energy.

For completeness, the fact that an object accelerates in a constant gravitational field with uniform acceleration is a consequence of galilean invariance, and the assumption that a gravitational field is frame invariant to uniform motions up and down with a steady velocity. Once you know that motion in constant gravity is constant acceleration, you know that

$$mv^2/2 + mgh = C$$

so that Huygens dynamical quantity which is additively conserved along with Archimedes mass times height is the velocity squared.

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By far the best answer. Note: the assumption $E(0) = 0$ should be made explicit. – Johannes Oct 28 '11 at 22:52

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That E(0)=0 follows from the fact that a stationary clay ball will not heat up the wall. – Ron Maimon Oct 29 '11 at 6:28

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+1 This is the answer. – becko Oct 29 '11 at 19:01

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@RonMaimon If you base the definition of kinetic energy on the concept of heat, how have to define heat first. How do you do that? – student Jan 29 '12 at 8:50

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@student: you can define it by the temperature change (as measured by volume expansion) of some reference material (mercury, for instance) for a large enough volume so that the change in temperature is infinitesimal. This is not an ideal practical definition--- you could use ideal gasses--- but for large volume the amount of heat flowing into a mercury thermometer can be accurately measured, and then you can calibrate your thermometers and measure specific heats. It's not a conceptually difficult thing, but it was historically difficult. – Ron Maimon Jan 29 '12 at 14:27
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The only real physical reason (which is not really a fully satisfying answer) is that $E \sim v^2$ is what experiments tell us. For example, gravitational potential energy on the Earth's surface is proportional to height, and if you drop an object, you can measure that the height it falls is proportional to the square of its speed. Thus, if energy is to be conserved, the kinetic energy has to be proportional to $v^2$.

Of course, you could question why gravitational potential energy is proportional to height, and once that was resolved, question why some other kind of energy is proportional to something else, and so on. At some point it becomes a philosophical question. The bottom line is, defining kinetic energy to be proportional to the square of the speed has turned out to make a useful theory. That's why we do it.

On the other hand, you could always say that if it were linear in velocity, it would be called momentum ;-)

P.S. It may be worth mentioning that kinetic energy is not exactly proportional to $v^2$. Special relativity gives us the following formula:

$K = mc^2\left(1 - 1/\sqrt{1 - v^2/c^2}\right)$

For low speeds, this is essentially equal to $mv^2/2$.

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The physical reason is, in some twisted way, Nöther's Theorem: the Energy is the conserved quantity with respect to time translations. And this can be calculated and shown to be the formula we all know and love. – Daniel Nov 11 '10 at 14:23

As Piotr suggested, accepting the definition of work $W=\mathbf{F}\cdot d\mathbf{x}$, it follows that the kinetic energy increases quadratically. Why? Because the force and the infinitesimal interval depend linearly on the velocity. Therefore, it is natural to think that if you multiply both quantities, you need to end up with something like $K v^{2}$, where $K$ is an 'arbitrary' constant.

A much more interesting question is why the Lagrangian depends on the velocity squared. Given the homogeneity of space, it can not contain explicitly $\mathbf{r}$ and given the homogeneity of time, it can not depend on the time. Also, since space is isotropic, the Lagrangian can not contain the velocity $\mathbf{v}$. Therefore, the next simplest choice should be that the Lagrangian must contain the velocity squared. I do think that the Lagrangian is more fundamental in nature than the other quantities, however, its derivation involves the definition of work or equivalently, energy. So probably you won't buy the idea that this last explanation is the true cause of having the kinetic energy increasing quadratically, although, I think it is much more satisfactory than the first explanation.

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+1: for going Lagrangian! – rubenvb Nov 11 '10 at 14:07
+1 for arguments of homogeneity and isotropy – P O'Conbhui Apr 21 '12 at 16:18
This can expanded by demanding that the Lagrangian give rise to invariant equations of motion under Galilean transformations. If you do this you find it must depend on v^2. – Mark Eichenlaub May 8 at 23:30

Let me just throw in an intuitive explanation. You could re-phrase your question as:

Why does velocity only increase as the square root of kinetic energy, not linearly?

Well, drop a ball from a height of 1 meter, and it has velocity v when it hits the ground.

Now, drop it from a height of 2 meters. Will it have a velocity of 2v when it hits the ground?

No, because it travels the second meter in a lot less time (because it's already moving), so it has less time to gain speed.

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In comes down to definitions.

Momentum is defined as $p = mv$. Momentum grows linearly with velocity making momentum a quantity that is intuitive to understand (the more momentum the harder an object is to stop). Kinetic energy is a less intuitive quantity associated with an object in motion. KE is assigned such that the instantaneous change in the KE yields the momentum of that object at any given time:

$\frac{dKE}{dv} = p$

A separate question one might ask is why do we care about this quantity? The answer is that in a system with no friction, the sum of the kinetic and potential energies of an object is conserved:

$\frac{d(KE + PE)}{dt} = 0$

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Actually, the momentum isn't equal to the time derivative of kinetic energy. – David Zaslavsky♦ Nov 11 '10 at 1:28
@David, you are obviously correct. Now I have to rethink my answer... – Ami Nov 11 '10 at 1:31
Looks a lot better now ;-) I wouldn't describe $d/dv$ as an instantaneous change, exactly, but that's semantics I suppose. +1 – David Zaslavsky♦ Nov 11 '10 at 1:39

One way to look at this question of yours is as follows:

$$E(v) = \frac{m v^2}{2} \; .$$

So, if we multiply the velocity by a certain quantity, i.e., if we scale the velocity, we get the following,

$$E(\lambda v) = \frac{m (\lambda v)^2}{2} = \lambda^2 \frac{m v^2}{2} = \lambda^2 E(v)\; .$$

That is, if you scale your velocity by a factor of $\lambda$, your Energy is scaled by a factor of $\lambda^2$ — this should answer your question (just plug in the numbers).

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Yes, but what is the physical reason behind energy increasing quadratically with velocity? – wrongusername Nov 11 '10 at 0:35
What's the physical reason that $F = m a$? Using this and a bit of differential calculus, it's possible to prove that the Energy is conserved (with respect to time derivatives) — and the formula for the Energy is this one you already know. – Daniel Nov 11 '10 at 14:21
– Daniel Nov 11 '10 at 16:11

For every relatively equal (in percents) increase of the speed, the applied force must be present over increasingly (quadratically) long travel distance. F=m*a. At the same time force*distance=work, where work=energy.

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I think it follows from the first law of Thermodynamics. It turns your definition of work into a conserved property called energy. If you define work in the $Fdx$ style (as James Joule did) then the quadratic expression for kinetic energy will follow with the symmetry arguments.

In his excellent answer, Ron Maimon cleverly suggests using heat to avoid a reference to work. To determine the number of calories he uses a thermometer. A perfect thermometer will measure $\partial{E}/\partial{S}$ so when he's done defining entropy, he still needs a non-mechanical definition of work. (In fact, I believe it is Joule's contribution to show that the calorie is a superfluous measure of energy.) The weakness in Ron's answer is that he also needs the second law of thermodynamics to answer the question.

To see this explicitly, write the first law in terms of the Gibbs equation: $$dE = TdS + vdp + Fdx$$ This equation defines $v = \partial{E}/\partial{p}$. For a conservative system set $dE=0$ and to follow Huygens, set $dS=0$ to get $vdp = - Fdx$ and to follow Maimon we set $dx=0$ to get $vdp = -TdS$. These are two ways of measuring kinetic energy.

Now to integrate. Huygens assumes $p$ is only a function of $v$. For small changes in $v$ we make the linear approximation $p = mv$, where $m \equiv dp/dv$. Plug that in, integrate, and you get the quadratic dependence. In fact, it's not too hard to see that if you use gravity for the force that $F = mg$ which leads to $$\frac{1}{2} m v^2 + mgh = C .$$ Raimon also has to assume the independence of $p$ on $S$. To integrate he will have to evaluate $T$ as a function of $S$ (and possibly $p$) or use the heat capacity.

Now notice that we required the changes in $v$ to be small. In fact, kinetic energy is not always proportional to $v^2$. If you go close to the speed of light the whole thing breaks down and for light itself there is no mass, but photons do have kinetic energy equal to $c p$ where $c$ is the speed of light. Therefore, it's better to think of kinetic energy as $$E_{kin} = \int v dp$$ and just carry out the integration to find the true dependence on $v$.

So, in summary, I suggest the "why" of the question is the same as the "why" of the first law.

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Imagine a cannon that fires a bed spring at 10 m/s (22 mph); the momentum of the fired bed spring carries it 25 meters. A second cannon fires the bed spring at 20 m/s (44 mph); the momentum of the spring carries it 50 meters. Doubling the speed, doubles the momentum.

In a second set of tests the first cannon fires the spring into an immovable brick wall from a distance of one meter. Upon impact the spring which has a length of 20 cm compresses to a length of 16 cm (20% compressed). When the second cannon fires the spring into the wall, traveling at twice the speed, the 24 cm spring compresses to a length of 4 cm (80% compressed); this of course requires 4 times more energy than the first collision with the wall. Double the speed, quadruple the kinetic energy!

Please correct me if I'm off base. thanks

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The general form of the kinetic energy includes higher order corrections due to relativity. The quadratic term is only a Newtonian approximation valid when velocities are low in comparison to the speed of light c.

There is another fundamental reason for which kinetic energy cannot depend linearly with the velocity. Kinetic energy is a scalar, velocity is a vector. Moreover, if the dependency was linear this would mean that the kinetic energy would vary by substituting $\mathbf{v}$ by $-\mathbf{v}$. I.e. the kinetic energy would depend of the orientation, which again makes no sense. The Newtonian quadratic dependence and the relativistic corrections $v^4$, $v^6$... satisfy both requirements: kinetic energy is a scalar and invariant to substituting $\mathbf{v}$ by $-\mathbf{v}$.

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Why can't it depend on the speed then? – Larry Harson Oct 13 '12 at 14:08
The kinetic energy for photons depends linearly on the speed. $E = |\mathbf{p}|c$. But this describes particles moving always at constant speed. – juanrga Oct 15 '12 at 10:06

Basically, momentum is related to force times time, and KE is related to force times distance. It is all a metter of frame of reference, either time or distance. The relationship between time and distance for a starting velocity of zero is $d = \frac{at^2}{2}= \frac{tV}{2}$. Plug this into the the equations you get the KE$= \frac{pV}{2} = \frac{p^2}{2m}$

Woolah - magic!

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Your intuition is NOT wrong and I will show you why. Say you have a machine that can accelerate a ball to 5 meters per second consistently. Put that machine on a car that is traveling at 5 meters per second and have it accelerate a ball. How fast is that ball traveling? The answer is 10 meters per second (5+5=10) as any physicist will tell you. If the machine is not moving, it changes the ball's velocity from rest to 5 m/s. When it is moving, it changes the ball's velocity from 5 to 10 m/s. The machine does not magically know that it is moving and so it always uses the same amount of energy, say electrical, whenever it accelerates the ball. If the kinetic energy formula were actually valid, that machine would not be able to cause the ball to accelerate to 10 m/s in this situation.

Those that tell you differently are only parroting the usual physics lessons. They can "prove" the kinetic energy formula is valid but you have to assume that work (the product of force and distance) is a proven fact; it is not. They might show an experiment that equates potential energy with kinetic that seems valid; this also depends on the assumption that work is a valid scientific concept. The example I gave in the first paragraph can be "explained" away by a clever physicist, maybe, but it is a direct test proving your intution is correct.
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