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- Hacker News
- Don't think about numbers double quadraple etc.
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
by G_o_D - It helps to re-frame the premise.
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
- >Energy is force times distance
This is not true though, work is only necessary to change KE. An object can have kinetic energy even when no forces are presently acting on it (of course force was needed to bring it from resting to that state though)
by krackers - Michael Spivak's Physics for Mathematicians has a lot of arguments like the one in the top answer here, answering questions about why the math of classical mechanics is the way it is.by aesthesia
- I didn’t think this was that weird. When you double your speed you are also going to be going twice as far in the same time, not just twice as fast, and they both have the effect of work.by hyperhello
- That made it trigger for me intuitively. Thanksby prism56
- To me the simplest way to understand it is through calculus. Kinetic energy is the integral of momentum so You go from p = mv to k = 1/2mv^2by tcoff91
- Mikes' answer is the most intuitive, but he rephrases the question in a possibly non intuitive way.
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.
by casey2 - This is basically it. If power is linear with respect to velocity, then it becomes illogical for kinetic energy to be linear with respect to velocity.
The energy of the object is simply the integral of power over time and that happens to be a quadratic function.
by imtringued - After reading a few answers I still feel like I haven't seen an intuitive answer to the question: why does it take so much more energy to go from 1 to 2 than from 0 to 1?
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.
by robalni - Sounds intuitive but what about rocket propulsion?by nakedneuron
- Cheat answer: velocity is a vector, and can be negative, while KE is a scalar and has to be positive. Therefore you have to square v to get rid of the minus sign.
Why not take the absolute value? Nature hates those, probably because the derivative is undefined at 0. So squaring it is.
by SyzygyRhythm - That's mnemonic not intuition.by johanyc
- > Why not take the absolute value? Nature hates those
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
- That doesn’t answer the title question of why it’s quadratic wrt speed.by qzw
- why not raise to any other even power ?by signa11
- I like to think of it as dot product being the true "natural" space to compare magnitude metrics, whereas absolute value is just a human construct conceived for our mental convenience. A smooth parabolic bowl vs an unnatural sharp conical tip. Also shows up in standard deviation etc.
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)
by xeonmc - Here's how to appreciate it in terms of the counterfactual:
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.
by Tazerenix - I love the counterfactual approach, thank you!!!
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!
by jmalicki - For me, the most intuitive explanation is that:
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.
- Yep. Momentum seems to be the source of our intuition. Something going 'twice as fast' has twice the momentum. OTOH, KE, being momentum * velocity, is more abstract.by 8bitsrule
- Ron Maimon uses an argument that relies purely on symmetry, which circumvents the standard explanations, including many in this thread. In some sense, this is the simplified version of Noether's theorem (as far as I understand it).
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.
by abetusk - It wasn't a permanent ban. He will be unbanned at Mar 18, 2292 at 16:28by Yajirobe
- It wasn’t “prescient”, StackExchange sites have always been among the most hostile communities on the Internet.
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.
by p-e-w - Fun little anecdote:
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
by throw0101a - heh, thats a fun little experiment.by senectus1
- Cool anecdote!
Couldn’t help but notice you misspelled car twice but only when talking about the blue car..
by slicktux - Very upset this didn't rely on doppler shifting of the car colors
- IIHS video shows the relationship between kinetic energy and speed in a very intuitive way:
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.
by linzhangrun - There's a great Australian traffic safety ad that makes this same point: https://www.youtube.com/watch?v=7x7c0qNGbv0by AlexandrB