(physics.stackexchange.com)
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.
What makes this intuitive? The foundation of the asker’s question is that it seems intuitive that kinetic energy would increase linearly with speed, but that turns out to be wrong.
I would not say we have the same intuition for kinetics. Increasing walking/running from 0 to 5 km/h doesn’t feel the same as than moving from 5 to 10, which does not feel the same as moving from 10 to 15. I don’t think we have an experience of linear relationship between running speed and effort, or other types of speed/energy types of relationships.
Getting up from a seat als walking a couple steps feels that same at home and in a flying airplane (or does it?). But the base speed is 0 in the former and several hundred mph in the latter case
Acceleration is a real force that we can feel. But once moving at a constant speed, physics dictates that it’s all the same. That’s also why you can throw a tennis ball up on a plane and not have it fly backwards immediately smacking into the person behind you.
In the reference frame of you and the aircraft, you are not moving at all and neither is the plane. In the reference frame of the ground you and the plane are moving.
and also pushing that reference frame down when moving up
And while the real situation at scale is more complicated, the math is going to come out to the same answer, albeit with extra terms muddying everything up.
If someone says that something true can be illustrated intuitively with a thought experiment, "sure, but what if we take that to a scale where our intuitions fail" is a sort of odd place to take the discussion unless you're genuinely curious how the math is going to shake out.
But more to the point the kinetic energy here is being turned into gravitational potential energy. If you move to a place with a weaker gradient in gravitational potential of course the same amount of kinetic energy moves you farther up.
You’re introducing two new intuitions, and it’s not intuitively obvious how they are related to each other. Why would work correlate 100% with caloric intake, and caloric intake 100% with kinetic energy?
Certainly, ‘work’ is highly counterintuitive. If I move a concrete block over loose sand on a beach, I’m doing zero work, in the physics definition, so moving it over a kilometer should be as easy as moving it for a millimeter.
Even ignoring the difference between caloric intake and caloric expenditure, it also isn’t intuitive to me that caloric expenditure is independent of the speed at which one lifts an object.
In the end, the answer is “because the math works out that way, and kinetic energy is a useful concept”
Stacking a weight on top of a table holds it at a fixed height and requires zero mechanical work.
The failure in intuition here relates to physiology and the mechanism by which muscles work, not physics. Myosin and actin are constantly cycling through bonding and release during muscle contraction, as this is how the shortening action actually occurs. In fact, muscle contraction is particularly unintuitive because people frequently consider ATP the "energy currency", yet the ATP-consuming steps are actually the release/relaxation and preparation for binding, not the pulling action. This is also why the phenomena of rigor mortis upon death occurs.
How I got banned from some reddit channel. Flip this around ask if a ball were fired out of a gun up into the air what height would it reach? A ball twice as fast goes up 4 times as high. If energy is force times distance it had 4 times the energy.
The journey from Y to Z might feel more tiring than the journey from A to B, but only if you do them all in one day :)
Not really, no. Not all forces are conservative.
But in the end, it's all up to the units/quantities we choose to measure, no? If we, say, decided to measure "Squenergy" in Sqoules, with 1Sq² = 1J, then suddenly, squenergy does increase linearly with speed! The formula for kinetic Squenergy becomes sqrt(m/2)v.
Of course this complicates other stuff, like potential Squenergy becoming sqrt(MgH), it not being additive, etc.
...no ? dropping something 10 times from 1ft is nowhere near energetic/damaging as once from 10tf
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
But if the cars produce downforce this is no longer true because you brake harder (more friction available) at higher speeds!
This is how F1 cars pull 4G when breaking. Some custom cars (like one of Ken Block’s last monsters or the Valkyre) use active aero braking to even greater effect.
2. I know you know this, but for the sake of others, it's when _braking_ (applying the brakes), not _breaking_ (becoming broken).
I'm not a pedant. But these errors jump out at me and I'm always a bit surprised and dismayed at this dichotomy; in our field, somehow the requisite attention to detail, the precision inherent to communicating scientific concepts, code, algorithms and formulae, is so often just abandoned when it comes to prose.
Honestly that was a typo and I noticed too late to edit. Thanks for catching
(I was suprised to see a cow jumping up on a ~3m rock ledge like it was nothing)
Point is that’s not always true. If they are the same type of car, and the car happens to be the kind with downforce, then their rate of deceleration greatly depends on air speed. A downforce car decelerates faster at higher speeds.
This is why you often see race cars lock their wheels towards the end of the braking zone, never at the beginning. The driver has to release the brakes as the car decelerates because there’s less friction available. You go from pulling 4G at the beginning of the braking zone to pulling the usual 1G once your speed drops enough for downforce to become negligible.
Alos! Many non-race cars actualy produce lift. Meaning the faster car decelerates at a slower rate than the slower car (0.8G vs 1G), making the effect from OP even more pronounced.
That’s not the only reason, and I’m not even sure it’s the majority reason.
Braking in a straight line offers more braking traction than braking while turning. What happens towards the end of a braking zone? The turn in. (Which also shifts weight to the outside tire and away from the inside tire.)
https://www.youtube.com/watch?v=RWwGFDynOHo
For these basic virtual car experiments, BeamNG.drive is a pretty good physics simulator. You can open its built-in tools and run braking tests directly.
5 km/h = 0.13 meter
30 km/h = 4.5 meter
60 km/h = 14 to 18 meter
65 km/h = 21 to 24 meter
You need 150% the distance at 65 vs 60.
In Dutch its remweg (something like brakeway) and my mind was occupied not finding the English word for it.
It cannot be both. It mathematically cannot be both. They can brake at the same rate (acceleration) or intensity (conversion of kinetic energy into heat) but because they are traveling different speeds those two values cannot be the same for both cars.
The math you did was for intensity, not force/acceleration, which because of the ^2 in the KE equation exaggerates the difference. Whereas if you did the math based on force you'd get a mild, linear, difference.
> and braked at the same rate,
You're being a bit sly with word choice here. You're doing the math for conversion of KE into heat whereas in common parlance "rate" means force/acceleration.
Braking "at the same rate" [of energy conversion] is way less actual braking force for the faster car.
This is basically the same kinetic energy into heat math wherein you can descend a grade at a low speed, apply a force and be fine and descend the same grade at a higher speed and apply the same force and cook the brakes. Or you can apply less force, and get the same amount of energy conversion into heat (i.e. your wording trick in the proposed scenario)
You've taken what's basically the math behind trucks descending a grade (rate of energy conversion is actually limited by ability of brakes to shed heat, not friction) and re-framed it as cars stopping to create a trick question.
You are right that the faster car is converting kinetic energy into heat faster per unit time. It also has less time to do so. The work formulation of the problem makes it obvious that these have to cancel out exactly.
Couldn’t help but notice you misspelled car twice but only when talking about the blue car..
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.
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.
Today they are additionally weighed down by increasingly erratic management decisions desperately trying to extract as much monetary value as possible before AI completely obsoletes SE, but the amount of aggression and hostility on the network was unbearable from the start.
I remember dozens of occasions where I looked up something on StackOverflow, intending to be in and out in 10 seconds, and ending up spending several minutes just staring in disbelief at the comments showing how people treat each other on that site.
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.
Back-of-napkin calculation says that if you managed to perfectly match exhaust speed with current speed, leaving all the expelled propellant stationary, it would only take quadratic amounts of fuels to reach higher speeds. Like the kinetic energy equation predicts.
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
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.
I've done plenty of this in pure math and stats, but this is the first time I've seen it applied to physics, and I love it! Thank you!
If I saw your derivation when I was 18 years old, who knows, maybe I would have caught the physics bug and went that way, this is super cool!
It's essentially the same argument: the Lagrangian can't have a bare a) position or b) velocity vector or it would violate homogeneity or isotropy of space, respectively.
Why not take the absolute value? Nature hates those, probably because the derivative is undefined at 0. So squaring it is.
Aside: I wonder if complex values neural networks with activation function just being sum(inputs)*conj(sum(inputs)) with threshold normalized by sqrt(num_inputs) could be the most universal, where incoherent inputs will average an absolute value of sqrt(N) and coherent inputs are N like lasers? (square amplitude would be N vs N^2 between uncorrected and correlated population)
For the purpose of inverting a negative vector, you can think of squaring as rotating the vector around the unit circle, 180 degrees, to make it positive. Higher order powers just keep rotating that vector back and forth- from this perspective the other even powers are the same transformation. Obviously with the magnitude being different.
And yet inverse distance laws for potential energy for gravity and electric fields use the absolute value because they require an unsigned distance and how you treat the singularity at zero is extremely important to the structure of those interactions
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
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
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).
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.
Not sure how I reconcile that for systems with linear symmetry that don't admit a sphere such as a 1D harmonic oscillator (i.e. a spring). You're confusing the fact that spheres require quadratics but quadratics are not sufficient to admit a sphere.
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.
∑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.
T = ½∑ⱼ mⱼ v⃗ⱼ²
Since space is isotropic, a Lagrangian can only depend on a speed vector through its norm. A Lagrangian must also be decomposable into independent orthogonal components, so you end up with an energy term that is shaped according to:
f(√(a^2 + b^2)) = f(a) + f(b)
Note: the components do not need to be independent and orthogonal for this to hold.
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).
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.
The energy of the object is simply the integral of power over time and that happens to be a quadratic function.
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?
I just felt like we never got to the heart of the matter of why the models work and how to approach developing them, it was all about learning a bag of tricks.
Meanwhile, math and CS being a lot more axiomatic by nature, they also made a lot more sense to me.
That being said, that specificity of physics, the unbridgeable gap between reality and the models we build to describe it, in retrospect, is what makes it more interesting to me today (it's not just a "closed" system in the sense that math is — of course the relationship between math and physics is itself fascinating but that's yet another topic), but I still feel like I haven't found the right pedagogical approach to make it fit my mindset.
Maths (and especially compsci!) are constructions by and for humans.
Is it any wonder it is as you describe? It would be odd if it was any other way.
Math and CS are mostly human-made, so most of the theorems/proofs/axioms are either straightforward or elegant—there are infinitely many possible axioms with no objective way to choose between them, so people generally choose to work with the ones that are the easiest for humans to reason about. You certainly could define a complicated set of axioms with dozens of special exceptions, but unless there are some external reasons why these axioms are important, nobody will want to work with them.
Conversely, physics exists to model the real world, so unlike math and CS, physics doesn't have the privilege of being able to choose the most convenient/elegant/simplest axioms to work with. Given the constraints of the real-world data, physicists will still choose the most elegant possible model, but sometimes a wacky model is the only way to accurately model the universe.
Nobody is really happy about this though, so physics textbook authors love to make their equations/derivations look simple/obvious/elegant, but this is often completely misleading, since often the rules of the universe are so weird that nobody would ever guess them without having ran the experiments first. But textbooks tend to downplay actual experiments in favour of post-hoc explanations, which tend to make the readers think that they're missing something.
> Physics is an endless source of frustration to me. It feels like a mix of random tricks, most of which I don’t understand.
Your feelings are correct, since physics really is mostly a set of random rules that nobody truly understands. But the important thing is that these random rules correctly model nearly everything in the universe to within a few hundredths of a percent, so they're not completely arbitrary.
> Are there good resources to learn it?
The annoying/inconvenient answer is to do lots of lab work. This is actually fairly accessible though, since a measuring tape, a scale, and a slow motion camera (present on any modern phone) is all that you need for most kinematics/mechanics experiments, and a multimeter, a 9V battery, some resistors, and some magnets are enough for most electromagnetics experiments.
Not sure if it'll help you with gaining an intuitive understanding, but at least it'll be interesting!
General advice take a look at the references in works you've recently read and look for lower level topics that interest you, after repeating a few times you'll find your way to physics or chemistry and you can use the above as reference works. The best resource is the one you actually use. If https://www.youtube.com/learning works better for you then use it.
The standard text to build understanding in physics is University Physics by Sears & Zemansky.
It's worth remembering you're quite far from the ground in physics, and it's mostly taught with "neat" cases that give insight into physics. I.e. the thought experiment to show why kinetic energy must scale quadratically with velocity is carefully designed to show that conclusion. You shouldn't expect to have come up with it off the cuff.
Energy is actually not a conserved quantity in Galilean relativity.
The answer linked above actually takes advantage of the fact that energy is not the same in different reference frames in order to make the argument work.
I think you are overthinking the heat thing. If you have a train car full of hot water and you slow the train down (extracting kinetic energy from it) until it stops, the water in the train car does not change temperature at all, other than a bit of sloshing around and loss of heat to the surroundings.
I don’t find the OP a convincing argument. What is temperature, why can you assume it didn’t change and the measurement also didn’t change commensurately? Why should kinetic energy be convertible with thermal energy? Chemical energy?
It’s very hand wavy and introduces many assumptions.
Kinetic energy is a book keeping trick. The real mystery is explaining how it relates to other forms of energy and how to tie it together.
a) energy is conserved in any frame of reference. b) energy can vary in 2 frame of references.
but then what it feels like is that when you reference the energy as mE(v), the v is actually not the only variable, and it will be more like mE(v, v_moving_reference)?
so we also must take intuitive that c) E(v, v_moving_reference) == E(v - v_moving_reference)
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.
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.
This account is temporarily suspended network-wide. The suspension period ends on Mar 18, 2292 at 16:28.
[1]: https://physics.stackexchange.com/users/4864/ron-maimon
A practiced arm with an axe beats a maul any day of the week. That's why splitting mauls are a modern device and splitting axes have existed since forever. Plenty of information on it online and on youtube, and why there are dozens of expensive specialty handmade splitting axes to buy and just cheap mauls for the rest.
Also this post is the physics behind it. Kinetic energy scales faster with speed than mass.
Splitting mauls are for people who either lack any experience or physically can't swing an axe that well. An axe is for people who got shit to do and don't have time to waste.
...
> 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.
So in some sense energy is momentum in the time direction (though it's not a Euclidean 4D space, so beware of assumptions). For an object at rest, this becomes its E=mc² equivalence. Kinetic energy is just a straightforward "rotation" of the frame.
However: Energy and momentum are not invariant under changes of reference frame, though the magnitude of the energy-momentum 4-vector is invariant between frames.
This is linear.
One small nuance... saying "kinetic energy is just a straightforward rotation of the frame" is close, but it's the total energy that is the time component of the four-momentum and mixes with the spatial momentum under Lorentz transformations. Kinetic energy is the difference between that transformed total energy and the invariant rest energy. So kinetic energy isn't itself a four-vector component, but it arises from how the time component changes when viewed from a different inertial frame.
Details about the specifics were hidden behind the scare quotes on "rotation". But sure, my phrasing was loose, how about 'What we ses as "kinetic energy" pops out of the Lorentz "rotations" of that energy in different reference frames.' ...?