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.
> 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
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.
1. +1 insightful, thanks for sharing your physics knowledge
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.
While it is true that some cars can brake harder due to downforce etc, the point from GP was that both cars brake/ decelerate at the same rate. Regardless of how exactly that deceleration is achieved.
> the point from GP was that both cars brake/ decelerate at the same rate
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.
> This is why you often see race cars lock their wheels towards the end of the braking zone, never at the beginning.
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.)
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.
OP wasn't explicit about taking the work = force * distance approach to dissipating energy. Two cars with the same mass and braking force (and thus deceleration) will put the same amount of work into the vehicle per unit distance, so will dissipate the same amount of energy in the braking maneuver.
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.
Nice bit of camera trickery. He says "both drivers react and a moment later they break", but the cars are still side by side. It (apparently) takes drivers 1.5 seconds to respond, the 5 km/h speed difference cuts the distance by 2 meter. Which apparently is a big deal. Rough estimate breaking distance:
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
The +5 km/h adds 6 to 7 meters or 8 to 9 if you account for response time.
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.
Physics is an endless source of frustration to me. It feels like a mix of random tricks, most of which I don’t understand.
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?
More than twenty years ago, I quit a program that taught math/cs/physics (the notorious French "classes préparatoires") ~almost precisely over this: I felt like I was being taught physics like it was an axiomatic system where the tricks should not be questioned, they just work so "shut up and calculate" (and you don't even need to be doing quantum mechanics for that).
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.
The world just is, regardless of what we think about it. Physics is our best attempt so far to understand and predict it at a low level, but it will always be incomplete.
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.
My point is precisely that I was often taught physics as if it was mathematics, where there is in fact a profound ontological difference between the two.
Your issue with physics but not with math reminds me a little of Hume's law. The difference that has always made that difference "make sense" to me is that math rules, even the axiom we use, are entirely chosen by the people using them, but the rules of physics are only useful if they match/predict what happens in the real world. For math we get to pick the ones that happen to be useful at a given time for a given problem (my go-to example of "it's all made up and the points don't matter" is why 1 isn't considered prime). For physics we're constrained to pick what best describes the real world. It probably helped that nearly all the physics course I had in high school/university had lab components focused on experimentally validating those rules/using those rules to predict results.
I think what it boils down to is that in my experience physics education lacks a clear historical component about how the current state of the art is a gradual process of proposing new models and rejecting old ones and figuring out the gaps between reality and the model. Instead, it feels like a God-given set of equations (that lots of people consider "the truth" for some reason), that you apply to cookie-cutter problems you must learn by rote. Though I understand the practical concerns (but then let's call it "physics for engineering"), as far as I'm concerned, you couldn't treat physics in a worse way.
What's the problem exactly? Could you not follow the example in the text?
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.
It seems that we're exact opposites! But if mathematics is your thing, it might be interesting for you to explore trying to learn things from a lagrangian perspective first?
Not sure if it'll help you with gaining an intuitive understanding, but at least it'll be interesting!
Lagrangian / Hamiltonian mechanics, the principle of least action, always seemed neat, in L&L and other places I encountered it, until I tried doing exactly what you're saying: gaining an intuitive understanding. At that point it just never made sense to me and seemed like a gratuitous deus ex machina that happens to work beautifully but for no apparent reason. You won't be surprised to learn I dropped out of my STEM program shortly after, though I keep a keen interest in the topic.
About the stationary action concept:
Yeah, it looks impenetrable, but here's the thing: there is a way of looking at it from just the right angle, and then becomes transparent.
Part of the story is this: the actual criterion is: the true trajectory corresponds to a point in variation space where the derivative of the action (derivative wrt applied variation) is zero.
In the cases examined when the concept was first introduced I suppose that in those cases the derivative-is-zero point was seen to be a minimum. From there, I suppose, came a supposition that there was some form of minimization at play.
However, within the scope of classical mechanics there are also classes of cases such that at the point in variation space corresponding to the true trajectory the action is at a maximum.
The above, and other aspects, are discussed in a resource that I created.
In the resource the mathematics is illustrated with interactive diagrams. Move sliders to sweep out variation. The diagram shows how the kinetic energy and the potential energy respond.
About interpretation:
As we know: motion along the true trajectory has the property that at every point in time the rate of change of kinetic energy matches the rate of change of potential energy. As we know: that property is known as the work-energy theorem.
The criterion derivative-wrt-variation-is-zero corresponds mathematically to the property: rate-of-change-of-kinetic-energy-matches-the-rate-of-change-of-potential-energy.
In the resource a two stage process is presented:
- Derivation of the work-energy theorem from F=ma
- Transformation from the work-energy theorem to classical mechanics stationary action
Of course: when you look at the work-energy theorem you wouldn't expect that it can be transformed to classical mechanics stationary action.
The transformation consists of multiple steps. In the resource I present it step by step; for each step the logic and consistency is readily recognizable.
For me, having the breakdown into mathematical elements available changed my whole perspective on classical mechanics stationary action.
I hope I can persuade you to check out the resource
Physics? Yes. Feynman Lectures On Physics and Computation. Landau & Lifshitz. If you like SICP you might like SICM. Nielsen & Chuang's Quantum Computation and Quantum Information then Faulkner's Modern Quantum Mechanics and Quantum Information
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.
Weird, I always loved physics because I felt like I didn't have to straight up memorize everything. In a pinch (years ago) I felt like I was able to pretty much derive everything I needed if I couldn't remember the exact formulas. It's all just forces and vectors.
Same for me. I wanted to major in physics and I quickly realized that I have no intuition for physics. Math made sense to me and I went to graduate school in math and still don’t understand anything in physics. Differential geometry, no problem. Electromagnetism makes no sense to me.
> 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.
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.
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.
Actually, it is momentum, sorta. Galilean 3D momentum isn't conserved under special relativity. The energy-momentum four-vector, however, is, under all lorentz-transformed frames.
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.
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.
To nitpick your nitpick: I know. But precision isn't the point here, it's to point out that there's an interesting and deeper symmetry at work. Energy and Momentum are not actually different quantities that vary in different ways but are still conserved via different laws. They're actually expressible as a single conserved vector quantity.
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.' ...?
If you use the right formula for calculating it (which approximates p=mv at low speeds), momentum is actually conserved in special relativity, and so is energy.
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.
Original comment is correct, it's not momentum. Work (hence, energy) is integral of force over distance, momentum is integral over time. There's not "sorta" about high school physics.
It doesn't make sense to me. Why split it into heat and motion and combine them to make 2 + 2 = 4 as if that solves the question? They are not the same units of energy.
This is also why splitting wood with a maul is way more work than using an axe. You can swing an axe at incredibly speeds which gives incredibly transfers of energy, but a maul is going to always have "meh" levels of speed because it is too much mass to accelerate over such a short distance as a swing. Also why you don't see framers using 3 lb hammers. You can put in more effort and get your lighter hammer swing to twice the normal speed, no way in hell you are doubling the speed of a 3 lb hammer though.
Ive split wood by hand my entire life to use for heat. Have you?
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.
sure have and I will take a maul over an axe. an axe is for felling not splitting. generally prefer a wedge and a sledge for splitting. gotta say some of the little electric splitters look sweet.
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.
One way of thinking about that is higher order even powers just reduce down to two.
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.
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)
> 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
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.
Energy is conserved in Galilean relativity. The thing you're trying to say is that it's not invariant across reference frames.
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.
thinking aloud here - so it seems like 2 things are taken as intuitive here:
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)
Yes that is what I meant. It’s not the same across reference frames.
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.
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.
It's easiest to visualize in terms of conversion from potential energy.
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.
Brilliant. For those wanting more numbers [0], the ball on the 10ft ladder hits the ground at (I'll stick with imperial units) 17.296 MPH, the ball on the 20ft ladder hits the ground at 24.46 MPH or 41.42% faster, and the ball on the 40ft ladder hits the ground at 34.59 MPH or 100% faster.
> We know intuitively that a ball atop a 20ft ladder has twice the potential energy of a ball atop a 10ft ladder.
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.
That's a good question, and I suppose the mgh formula isn't a suitable answer, so my answer would be something like: if you lift an object to some height, and then you repeat that action (lifting it from there to twice the height), you've done twice the work, and doing twice the work requires twice the caloric intake.
Okay but that depends on the intuitions the question is trying to justify, which makes it circular. We also know, for example, that the body uses more than twice as much energy to do twice as much work (because of fatigue on the muscles or whatever the right term is here). In fact it takes positive energy just told a weight at a fixed height, doing zero mechanical work! So you’re actually appealing to even weaker intuition than the one the question is trying to ground!
What point that I made are you responding to? I was disputing someone’s appeal to a specific intuition for being an unhelpful one to use here. So I obviously get the concept of some intuitions being useful. Did you see the comment being responded to?
I think if you define energy as force X distance then integration alone will give you the squared term.
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.
> In fact it takes positive energy just told a weight at a fixed height, doing zero mechanical work!
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.
I get that involving a human body complicates the analysis. That was the point: that you can’t appeal to it as a simple example to ground the intuition in other case.
Also:
>Stacking a weight on top of a table holds it at a fixed height and requires zero mechanical work.
I was referring to a human holding it. What would have been a better way to keep you from missing that?
> I get that involving a human body complicates the analysis. That was the point: that you can’t appeal to it as a simple example to ground the intuition in other case.
Yeah, fair enough. It's unfortunate that the comment you were responding to involved "caloric intake" suggestive of a biological system when it could just as easily have involved a mechanical pulley. Their intuition would have been stronger phrased as: Within a gravitational field that is approximated as a uniform force field, the amount of energy/work to raise a weight by a fixed distance is independent of the initial position of that weight on a (massless) rope attached to a (frictionless) pulley.
> I was referring to a human holding it. What would have been a better way to keep you from missing that?
Since you are asking, for me, it would helped to have the following inserted text: "In fact it takes positive energy [for biological muscle] just told a weight at a fixed height, doing zero mechanical work!" so that the statement stood on it's own, or else including within your post a more definitive statement to the effect of "intuitions involving the human body complicates the analysis", as you did in this reply. If "that was the point", go ahead and state it.
To be fair, I often have the same problem. It's easy for me to write a lot of text that goes around a statement without actually getting to it.
> if you lift an object to some height, and then you repeat that action (lifting it from there to twice the height), you've done twice the work, and doing twice the work requires twice the caloric intake.
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”
I think you're missing the point. A lot of basic mechanics actually isn't especially intuitive because things like work simply do not map well to everyday experience. I'm not suggesting that work is defined incorrectly or something.
"Work" is the weird thing in physics, I'd say is about as opposite to intuitive as you can get when introducing a concept. It's only intuitive when considering lifting an object - say, a bag of groceries. Heavier the bag, higher the lift -> more work. But then you carry that heavy bag a couple kilometers, arrive at home exhausted, only to be told by the physics teacher that you did exactly 0 work. Or in fact negative work, if upon coming home, you put the bag down.
I understand the concept myself somewhat intuitively now, but that intuition is not connected to everyday experience - it's just familiarity with a detached concept of physics!work that just is what it is, but is consistent in being that.
Because things like energy are relative. So if you label the ground 0, and go up 10 feet, you get x energy. Going up another exact same x from your 10 foot ladder spot you could now call 0 again, would mean you gain x energy again.
Since they're both the same height, and you gained the same energy, you could infer double the height has double energy.
What if you label standing still as 0 mph and start moving 10 mph, gaining x energy, then call that zero and start moving 10 mph from there? It's just as intuitive to say that you would gain x energy in that case, but you don't.
When you're already going 10 mph and you're about to add another 10 mph, you can only "call that zero" (i.e., go from 0 mph to 10 mph again) if your point of reference (i.e., the ground) also begins moving with you at that point. Since the ground is stationary, you're definitely about to increase from 10 mph to 20 mph relative to the ground, not from 0 mph to 10 mph, and that's harder to do. But if you're on a treadmill that was stationary for the first change, and then suddenly starts moving at 10 mph right before the second change without affecting your speed relative to the ground, then you can "call that zero" and you'll be able to add another 10 mph (ending up at 10 mph relative to the treadmill and 20 mph relative to the ground) with the same ease as the first go.
I suppose they are both "intuitive", but the example I gave was both intuitive and correct. Probably for anyone who has carried something or themselves up a hill, or climbed a set of stairs can relate to that from firsthand experience. I don't know what the kinetic energy corollary to that would be? "Stand still and I will throw a baseball at you going 15mph, and note how much it hurts. Now I will throw it at you going 30mph. See! It hurts 4x as much" :D
That's clever, and I can't imagine or explain it as easily.
Something to do with a reference point moving away from you so when solving for bringing it back to zero it's different than just adding the two energies back together. You have to add up the energy of catching them all up to the initial starting reference.
I think also because distance is one unit, so moving reference pointe is easier. Moving reference points on distance over time already gets my spidey senses going that it's not something you should do without some real understanding.
Then 20ft should not be used in the explanation. They should just have one ball going at 2x speed hit the seesaw and have 4 of those balls go up at 1x speed.
That ends up begging the question, because the next step is "how high do you have to drop it from so that it's travelling twice as fast?" and you're immediately going round in circles.
The effort to move a piece of furniture from 1st to 2nd floor is the same as the effort to move it from the 2nd to the 3rd. We have good intuition for this by our experience, which derives a linear relationship. The effort to move a piece of furniture up two floors is double the effort of moving it up one floor (ie you have to put the same effort twice, assuming enough rest).
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
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
Couldn’t help but notice you misspelled car twice but only when talking about the blue car..
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.
(I was suprised to see a cow jumping up on a ~3m rock ledge like it was nothing)
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
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.)
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.
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.
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.
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.
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.
Not sure if it'll help you with gaining an intuitive understanding, but at least it'll be interesting!
https://en.wikipedia.org/wiki/Lagrangian_mechanics
Part of the story is this: the actual criterion is: the true trajectory corresponds to a point in variation space where the derivative of the action (derivative wrt applied variation) is zero.
In the cases examined when the concept was first introduced I suppose that in those cases the derivative-is-zero point was seen to be a minimum. From there, I suppose, came a supposition that there was some form of minimization at play.
However, within the scope of classical mechanics there are also classes of cases such that at the point in variation space corresponding to the true trajectory the action is at a maximum.
The above, and other aspects, are discussed in a resource that I created.
https://cleonis.nl/physics/phys256/energy_position_equation....
In the resource the mathematics is illustrated with interactive diagrams. Move sliders to sweep out variation. The diagram shows how the kinetic energy and the potential energy respond.
About interpretation: As we know: motion along the true trajectory has the property that at every point in time the rate of change of kinetic energy matches the rate of change of potential energy. As we know: that property is known as the work-energy theorem.
The criterion derivative-wrt-variation-is-zero corresponds mathematically to the property: rate-of-change-of-kinetic-energy-matches-the-rate-of-change-of-potential-energy.
In the resource a two stage process is presented:
- Derivation of the work-energy theorem from F=ma
- Transformation from the work-energy theorem to classical mechanics stationary action
Of course: when you look at the work-energy theorem you wouldn't expect that it can be transformed to classical mechanics stationary action. The transformation consists of multiple steps. In the resource I present it step by step; for each step the logic and consistency is readily recognizable.
For me, having the breakdown into mathematical elements available changed my whole perspective on classical mechanics stationary action.
I hope I can persuade you to check out the resource
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.
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.
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.
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.
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.' ...?
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.
[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.
Why not take the absolute value? Nature hates those, probably because the derivative is undefined at 0. So squaring it is.
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.
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)
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
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.
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)
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.
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.
[0] https://www.omnicalculator.com/physics/free-fall
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.
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.
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.
Also:
>Stacking a weight on top of a table holds it at a fixed height and requires zero mechanical work.
I was referring to a human holding it. What would have been a better way to keep you from missing that?
Yeah, fair enough. It's unfortunate that the comment you were responding to involved "caloric intake" suggestive of a biological system when it could just as easily have involved a mechanical pulley. Their intuition would have been stronger phrased as: Within a gravitational field that is approximated as a uniform force field, the amount of energy/work to raise a weight by a fixed distance is independent of the initial position of that weight on a (massless) rope attached to a (frictionless) pulley.
> I was referring to a human holding it. What would have been a better way to keep you from missing that?
Since you are asking, for me, it would helped to have the following inserted text: "In fact it takes positive energy [for biological muscle] just told a weight at a fixed height, doing zero mechanical work!" so that the statement stood on it's own, or else including within your post a more definitive statement to the effect of "intuitions involving the human body complicates the analysis", as you did in this reply. If "that was the point", go ahead and state it.
To be fair, I often have the same problem. It's easy for me to write a lot of text that goes around a statement without actually getting to it.
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”
The fact that your muscles burn ATP just fighting gravity is a feature of biology, not fundamental to the physics involved.
If you want to read about another similar example, Google for rocket launch gravity losses.
I understand the concept myself somewhat intuitively now, but that intuition is not connected to everyday experience - it's just familiarity with a detached concept of physics!work that just is what it is, but is consistent in being that.
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.
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