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 even close to twice the speed at impact. So let's look at why not.

The force of gravity is a constant force that causes constant acceleration in free fall regardless of speed. (Ignoring air resistance, inverse sq considerations, etc.)

Suppose it takes 1 second for the ball on the 10ft ladder to hit the ground with kinetic energy of 10 and a speed of 100. Again, gravity as a constant acceleration force is speed increase per time... not speed per distance. In the ladder example, it took 1 full second for gravity to accelerate the object to speed 100.

Now think about the 20ft ladder: the ball is dropped. How much kinetic energy and speed does the ball have after it has fallen 10 feet (but still has 10 left to go)? Well it has the same exact amount as the other ball did after falling 10 feet for a duration of 1 second: kinetic energy of 10 and speed of 100.

Now the crux: thinking about when the final 10 feet of the fall look like. We know for sure the ball still has 10 ft of potential energy to covert into kinetic, and that that will happen as it falls. But what of the impact speed? Since the current velocity of the ball as it enters the last 10 feet is already 100, we know it will spend less time transiting this distance than it did the first half where it started at off at speed 0. Since gravity imparts speed in free fall as a function of time - consequently less speed will be imparted over the second 10 foot interval. That concept is enough to prove the relationship isn't linear.

If you do the actual calculation or tests, you will see one ball needs to be dropped from 4x the hight of another to hit the ground at 2x the speed, but yet with still 4x the kinetic energy.

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 rate, it doesn't managae to stop: but what speed is it going when it hits the obstacle?

The blue car, using ½mv², shed (~70²=) 4900 units of energy (we'll hand wave away the constants). So the red car, which had (100²=) 10000 units of kinetic energy to start, also shed 4900 units, which means it had 5100 units of energy when it collided, and so was going (√5100~) 71.

* Numberphile: https://www.youtube.com/watch?v=i3D7XYQExt0

As an aside, I believe Ron Maimon's account was suspended after he challenged the character of someone who was soliciting votes for a moderator position. Ron Maimon's stance was that if someone was running for an elected position, discussing their character was valid. The SO site had/has a strict challenge-the-question-not-the-person policy, which the moderators used to ban him permanently.

At the time, I remember seeing some posts by Ron talking about how the SO sites were corrupted by their policies and that it was a matter of time before they ceased to provide value. I think this was late 2000s or early 2010s. Looking back it's hard not to feel like his stance was prescient.

Force = change in momentum with time

Energy = Force x distance

Now consider how much energy can be dissipated by a tiny change in momentum over a small distance dx, when we are at a given velocity v:

dE = Fdx = (dp/dt)dx = m(dv/dt)dx = mdv(dx/dt) = mv*dv

The intuition is that in order to apply a force through some distance, I have to change the velocity of an object by dv. But, the distance I just traveled also depends on the current velocity v. That's why the total energy available isn't just simply proportional to velocity - every time we change v, the amount of force available goes down, too.

Summing all the little bits of energy dE over our velocity changes dv, from the starting velocity down to zero, and we get the formula for kinetic energy.

BTW, the intuition here really starts from the idea that force = momentum change with time. The definition of "force", "momentum", and "energy" can be maddeningly circular, even if we have clear mathematical representations and a common world we experience.

Suppose kinetic energy was E = m|v| instead, linearly dependent on speed |v|. What does that mean for the universe?

The traditional Lagrangian is L = 1/2 mv^2 - V(x). This kinetic energy gives a different formula:

L = m|v|ln|v|-V(x).

Deriving the corresponding equations of motion, you get:

p = m(1+ln|v|)sgn(v)

ma = |v|F

A few things we can note from these formulas:

1. They are not boost invariant: Galilean relativity is violated. That means there is necessarily a privileged reference frame (i.e. an aether) in which the universe is at rest, and all dynamics must be understood relative to this reference frame.

2. Newton's first law has a pathological interpretation in regards to the above reference frame: If ma = |v|F and |v| = 0 (i.e. you are at rest relative to the aether), then a = 0 no matter what F is. That is, for objects which are stationary with respect to the aether, no motion is possible regardless of what force is applied.

It is still true that objects in motion (relative to the aether) remain in motion unless acted upon by an outside force, and Newton's third law is still true, but such a universe basically makes no sense.

You could essentially argue from the anthropic principle that such a universe would have such pathological dynamics that it could not permit life, and therefore we cannot observe it.

This is the contrapositive of the argument presented on stackexchange. There they say "given Galilean relativity, you get the quadratic scaling law". This argument says "if you don't have the quadratic scaling law, you don't have relativity".

The point of the counterfactual is a bit like Richard Feynman's "why" argument [1]. There is no fundamental reason why this kind of dynamics couldn't exist. We can only ever reduce our explanation to a more fundamental intuition we have about the same universe we live in (i.e. from kinetic energy scaling laws to Galilean relativity). But without a mathematical proof of the incoherence even in principle of the alternative, its perfectly valid to imagine an alternative universe with different dynamics. It's just not our universe.

[1] https://www.youtube.com/watch?v=36GT2zI8lVA

Why not take the absolute value? Nature hates those, probably because the derivative is undefined at 0. So squaring it is.

I have been thinking about it and only been able to come up with something that feels intuitive but not at all precise and I don't know how correct.

When you stand still you may use your surroundings to gain some speed, like by pushing against a wall.

When you have speed it gets harder to gain more speed because the surroundings are (relative to you) moving in the wrong direction, so for every additional unit of speed, it takes more effort to get there.

Odd that nobody mentioned power, which scales linearly with speed. Of course if you add linear increasing amounts of power to the system the energy will increase quadratically.

Power scaling linearly is more intuitive because doubling your speed requires twice the power to maintain the same force, why does it require twice the power? because you have half the time to power it.

An object which has a constant force applied will have it's distance increase quadratically with respect to time.

Energy is force times distance. Intuition: the energy it takes to lift an object up is proportional to the height you lift it to.

So if you apply a constant force, you get a constant acceleration which leads to a quadratically increasing distance.

If you accept that energy is force times distance, the energy required to move the object in this scenario increases quadratically.

This means that if you apply a force F for 1 second, the amount of energy that is imparted by that force depends on how fast the object is already going. The energy required to apply a force to an already fast moving object is much higher. Intuition: you have to expend all the energy required to get up to the moving object's speed before you can start applying a force. So there's a cost to even get in the game

Think of simple notion. Why more energy is needed to accelerate moving object compared to still?

Kinetic energy possesed by any object is equal to work/effort needed to be made by an external force to accelerate it from present state to stated velocity.

If object is already moving, and i am that external force, first i had to catch up with that object, for that i had to do work make effort until i am moving at same speed as object, even after catching up, at the momennt if i try to push object, i am distracting myself engaging into 2 activity maintaining my speed same as object and trying to push so that will definetely reduce my speed, so i first had to gain slighly more speed than object before i give it a push and transfer all my momentum to object so it accelerates.

Thus i needed more effort or work to do, to accelerate moving object compared to stationary one.

That work done is kinetic energy object posses when it was accelerated from 1 to 2 and its more than when moving object from 0 to 1.

That simply explains the fact. Now how much more energy triple or quadraple that comes down to practical established formulas.

In my understanding OP was confused as when talking about,op was simply thinking if object is already moving it would take less force to move it as it already has gain momentum against all odd of nature and resistive forces, so now only work needed is to accelerate it and it doesn't include loss against resistive forces.

But to accelerate moving object the applier of force whether human or another object also needs to catch up

There are often two ways to solve physics problems: one describing the problem with forces, the other reasoning with energy. So they look like the two faces of the same coin. Hence the question: which one is actually real?

Some quick arguments for and against

Energy:

+ converts between mechanical, chemical, thermal, radiative types, and even mass

+ quantum particles, when interacting, exchange energy

- looks like an integrative quantity (in the sense of mathematical integral)

Force:

+ feels very real, when you receive a ball in your face

+ we talk of fundamental forces, not fundamental energy

+ explains momentum, deformation well

- my physics teacher used to say "nobody ever saw a force"

- force is undistinguishable with acceleration

- at the quantum level forces are actually particles interacting

- at the quantum level, the uncertainty principle makes the newtonian force pointless (pun?): seems like we could know the vector's origin or the direction but not both

∑ⱼ mⱼ v⃗ⱼ = 0⃗

where the mⱼ are the masses of the parts of the object and the v⃗ⱼ are the velocities of those parts.

If the object initially has 0 velocity, its kinetic energy is:

T = ½∑ⱼ mⱼ v⃗ⱼ²

Now we give the object a kick (or just switch reference frames) to change its velocity by Δv⃗. The new kinetic energy is:

T' = ½∑ⱼ mⱼ (v⃗ⱼ + Δv⃗)²

T' = ½∑ⱼ mⱼ (v⃗ⱼ² + 2v⃗ⱼ⋅Δv⃗ + Δv⃗²)

T' = ½(∑ⱼ mⱼ v⃗ⱼ²) + Δv⃗⋅(∑ⱼ mⱼ v⃗ⱼ) + ½Δv⃗²(∑ⱼ mⱼ)

If M is the total mass of the object, then we can substitute this into the sum in the last term. And we already saw that the sum in the middle term was 0. So:

T' = ½(∑ⱼ mⱼ v⃗ⱼ²) + Δv⃗⋅0⃗ + ½Δv⃗² M

T' = ½∑ⱼ mⱼ v⃗ⱼ² + ½MΔv⃗²

So in terms of the original kinetic energy T, which was purely thermal energy, we get:

T' = T + ½MΔv⃗²

In other words, because of the quadratic kinetic energy formula, we can see that the total kinetic energy T' of a hot object is just its thermal kinetic energy T plus the usual mechanical kinetic energy ½MΔv⃗².

For instance we know that the life forms grow via cell division, but no text can address the question of "why". They can only talk about "how".

Infact, science quest is not really about answering "why" all the way down the causal chain. It is about learning how the qualities of things are related and a bit of shallow causal chain inspection.

The causal chain, by nature, does not allow full inspection. It's dependency on temporal constructs means it breaks down where time breaks down. Infact causality might might break down at macro levels as well, leading to loops with no end or beginning (kind of chicken and egg problem).

Assume you have a fan sitting still. You smack it and it’s now rotating with 1m/s angular velocity. If you want it to go faster you can’t smack it at the same speed. You have to hit it faster else you’re just tickling it and it stays the same speed. So you smack your hand twice as hard and now it’s going even faster. Then three times as hard, four times, etc.

If you sum the smack energy it will be 1+2+3+4, which starts to build out a right isosceles triangle if you graph it. Such a triangle is half of a square, ie: 1/2*v^2.

If one starts with Newton's 2nd law (F=ma) assumed, then one can derive kinetic energy to be 0.5mv^2, and this is what most of the answers are explicitly or tacitly doing.

One could however start with Lagrangian formulation along with KE = 0.5mv^2 and drive F=ma. This is where one needs an explanation for why KE = 0.5mv^2, and the first answer (@Ron Maimon) is providing an explanation.

Most books I have come across on Lagrangian formulation secretly assume Newton's laws.

In my opinion, Lagrangian formulation can proceed without Newton's and without even defining momentum as mv, however, now needs KE = 0.5mv^2.

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.

Velocity is itself an expression of energy, and energy does scale linearly. Why does it take quardically more energy for each extra unit velocity? Well, because the closer you get to the speed of light the harder it is to get closer still. Why that is simply requires a deeper appreciation of the system than I expect is possible to muster in a comment.

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.

But as others have mentioned this is only as intuitive as F=ma, or p=mv.

In my view, at least classically it's just a matter of definitions then. If our definitions of energy differ, the only thing we will experimentally agree on is the equation of motion, and even then up to a frame transformation.

Kinetic energy is the work needed to accelerate something from rest to speed v (velocity). Below we'll use the regular derivative representation to represent an tiny change in a value using notation such as dE (tiny change in energy).

Work can be described as dE (tiny change of energy) = F (force) * dx (tiny change in position). -> dE = F * dx

Force though changes momentum and algebraically can be described as:

F (force) = dp/dt (tiny change in momentum over tiny change in time)

The distance travelled over a infinitesimally small time frame is:

dx (tiny change in position) = v (velocity) * dt (tiny change in time) -> dx = v * dt

So now we have:

dE = F * dx (see our definition above) = dp/dt * v * dt (due to F = dp/dt and x = vt) = v dp (the dt terms cancel each other out and we are left with velocity and rate of change in momentum) -> dE = v * dp

So we are therefore left with:

dE (tiny change in energy) = v (velocity) * dp (tiny change in momentum) -> dE = v * dp

We can translate this as stating: the energy cost of adding a tiny bit of momentum depends on how fast the object is already moving. Now we also know that algebraically:

p (momentum) = m (mass) * v (velocity) -> p = mv

Using derivatives we also have:

dp (tiny change in momentum) = m (mass) * dv (tiny change in velocity) -> dp = m * dv

Going back to our original equation, we now have:

dE (tiny change in energy) = v * dp = v * m * dv (since dp = m * dv)

To get the total energy that it takes to move a mass m from 0 to velocity v, we need to add up all the tiny energy costs from 0 to v, and for this we use the integral to get:

E = ∫(from 0 to v) add up m (mass) * v (velocity) * dv (rate of change in velocity) = 1/2mvv (basic integration) = 1/2m*v^2

So from the above we can see that the kinetic energy rises quadratically with velocty. Now the 'why' this is encoded in the universe: read a bit more about symmetry and Emmy Noether. The laws of physics do not care where you are, when you are, or whether you are moving at a constant speed in a straight line. This is called Galilean symmetry in ordinary classical mechanics. Because the laws of motion have to stay consistent under changes of reference frame, energy cannot just be proportional to velocity. A linear energy law would break the symmetry.

gravity will accelerate a ball. this is not a linear process. the heat generated by collision with the ground is not double, but quadruple.

so the only thing that is linear is the DISTANCE.

Define (a)work = energy, (b)work = force x distance and (c)force = mass x accel. Substitute c into b you get work = distance x mass x accel and substitue into a you get energy = distance x mass x accel.

By this equation, an apple falling twice the distance, (and having a constant mass and acceleration) will only have twice the energy.

This 'lie' of quadratic energy growth is just another magic trick physicists have used to confuse students.

Or someone runs by and you want to push him in the back to go faster.

You will have to push with great vigor, unless you first get up to speed yourself (also takes energy).

I find math and compsci reasonably understandable, can read research papers in both fields ( and have published papers) etc. There’s something specific about physics I don’t get but I’ve never been able to figure out what. The main symptom is that most cause -> consequence in such demonstrations , which are seemingly obvious to everyone, make no sense to me.

Am I the only one ? Are there good resources to learn it?

Energy is actually not a conserved quantity in Galilean relativity.

If I'm remembering correctly, this is also why the energy required to "reach" the speed of light for subjects with mass parabolically goes to infinity. I'm also guessing it can directly trace a proof down to why kinetic energy increases quadratically.

...

> 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.

That's still just pushing the problem elsewhere. Intuitively, why does the two-ball system end up with a velocity of 1v?

F=ma (Force equals mass times acceleration)

W=Fd (work equals force multiplied by distance)

V^2=2ad (velocity squared equals two times acceleration times distance)

So W = Fd = ma(v^2/2a)

Finally: W=1/2mv^2 (work equals 1/2 mass times velocity squared)

So this explains why car crashes can be so dramatic, as a doubling of speed results in 4x the kinetic energy.