minutephysics - 2014-12-20

WATCH MY VIDEO ABOUT AIRPLANES! https://www.youtube.com/watch?v=7rMgpExA4kM http://www.audible.com/minutephysics MinutePhysics is on Google+ - http://bit.ly/qzEwc6 And facebook - http://facebook.com/minutephysics And twitter - @minutephysics Minute Physics provides an energetic and entertaining view of old and new problems in physics -- all in a minute! Music by Nathaniel Schroeder http://www.soundcloud.com/drschroeder Created by Henry Reich

I'm sensing a pattern here "I don't understand, therefore video is stupid"... Really internet people?

I think the main issue is that he talks really quickly.

I thought KE was velocity. :/

But I still don't understand, why didn't minute physics apply a bit of calculus? That is more fun and simpler.

Technical people usually produce poor presentations as a rule.

You just explained the internet

"The video was too quick", the channel is called "minute physics". Pause, reverse, you can do that.

Gonzalo Ayala Ibarre Are you talking about the length of the video or the content/length ratio of the video?

The calculus proof at the end was much easier to understand

+Ray Louis And it took 20 seconds

+Ray Louis

There is something. . different about your profile picture. I just can't quite put my finger on it.

100% agree. More precise, more understandable, and it took only 20 seconds

Such is the power of math

I assure you all that I completely understand what I just watched and how it can be used, and by no means was completely confused and stared at my screen with a blank expression the entire time.

Regardless, enjoyable video for sure.

Nils Nilsson same here

Me too

Nine Eleven Also is not true that energy is conserved. Theres no such thing as "conservation of energy" in the universe. The expansion of the universe is an example. Physics should always explain it because everynody seems to be confused with this concept. This is 200% bs. Energy is not conserved at universal scale.

O PALMEIRAS conservation of energy is a fundamental special case of the Theorem of Work Energy. If there’s no other external work acting on the system, then the energy theorem reduces to conservation of energy. If there’s an external work affecting universal scale, then your argument might have a valid point. Regardless of, the explanation in this video is an amazing algebraic simplification of the differential analysis of energy and it should be noted.

Nils Nilsson agree, this was a beautiful simplification of the calculus prove of conservation of energy.

i understand the conservation of energy part of it, but where do the time zones come into it?!

ah ok now i get it! thanks.

+King0097 The timezone thing is a pretty abstract idea. It is basically, since there isn't anything special about time (I mean, what time is now, compared to what time we can set on the clock) we can choose our time by convenience. If there was any particular asymmetry about how we can choose the time on a clock, we wouldn't be able to freely choose the time on our clocks. It's a pretty abstract idea, that also leads to the conservation of energy.

If there was absolute time for all places on earth then it means that conservation of energy does not exist. Variable time is important for conservation of energy. Varying time is the one that predicts conservation of energy in the first place.

I'm Still confused about the timezone thing.

Tangents and cosines. (sorry I went off on a tangent *badum crash*)

I enjoyed the music though

This video made so much more sense after I started taking physics

Can't believe I watched this so many times without noticing that extra calc bit at the end, that was really cool are there more minutephysics vids with that? :-)

Very end of it, he did it fast so pausing was necessary but I greatly appreciated it :D

Yeah, I looked over it again as per your claim. No calculus.

@Ho Lee Fuk its after the credits, around 3:20

Oh, whoops. I assumed the video ended when the credits rolled. My apologies :-)

@Ho Lee Fuk it's cool lol i didn't notice it either till one day when I was too lazy to click away and just let it roll

yeah... very simple...

Yup

Alex Stefanov first semester physics, bud

I thought this was a SIMPLE proof.

It is compared to Lagrangian mechanics

It is simple

Charlie Snowball simple in physics isn't simple

It is extremely simple

It's simple when not explained in two minutes

I was GOING to take physics during my next semester of college... but now Im thinking ill just take something else.

Nah. This is kindergarten physics.

Naaah, It's pregnantgarten physics.

Naaaaah, I knew this stuff in the womb

naaaaaah, I knew this stuff is a sperm

Pretty simple once ya think abou tit

Ikr.

This is a circular proof; the kinetic energy formula (1/2 m v^2) is made so that it equals the work the force vector sum do on the object. The formula is derived form conservation of energy in the first place. Richard Feynman does a better "proof" in The Feynman Lectures on Physics. See the introduction to energy section.

@NeedsEvidence Aren't the the laws of mechanics invariant under time translation by their formulations in the first place? If, for example Newton's second law WAS time dependent and acceleration was inverse proportional to the age of the universe, wouldn't the time dependence have to be stated? Wouldn't Newton's second law had been something like this: (sigma) F = k*t*m*a were k is a constant and t is the age of the universe.

My problem with this video is that it tries to prove that a quantity - which comes in two forms, differing only by a minus sign - is conserved by adding the difference over time of each of the forms together. Of course they will equal zero, since something added together with minus one times itself is nothing. (You could also "prove" that the total velocity of an object is conserved by defining a form of velocity called phantom velocity, which equals minus "normal" velocity. Add the difference in "normal" velocity over time with the difference in phantom velocity over time and you get zero. The total velocity is the same! :D Q.E.D!) We need to know that kinetic and potential energy are two forms of the same quantity, a quantity that actually exists and enables an object to do work.

It is essential to a proof of conservation of energy to first prove that the work an object can do by reducing its speed to zero (i.e kinetic energy) is 1/2 m v² (or simply the work done on the object when taking its speed from zero to v). One must also prove that the work an object can do by moving in a force field with the force (i.e potential energy) is -W (the work done on the object when moved it against the force). All the proofs I had seen of these formulas assume conservation of energy, until recently. I found out that you can prove the formulas using Newton's third law and the definition of definite integrals. When doing so, it wasn't hard to see that the work an object can do by reducing its speed to zero is the same work done on the object when taking its speed from zero to v. It was just as easy to see that the work an object can do by moving in a force field with the force is the same (but negative) work done on the object when moving it against the force. I have proved myself wrong, Henry's proof is not circular. But sadly he doesn't explain why it isn't in the video himself.

@iamihop > I find Wikipedia to be a poor teacher, packed with jargon and novel symbols in cascades of equations.

You are right, the link is of no pedagogical use. It was meant to illustrate the "more robust proof" mentioned at the end of the video: 1) There is another starting point to mechanics which is not Newton's laws of motion but a quantity called the "action" (the expression S in the link) that is **axiomatically build from kinetic and potential energies** (the expression in brackets is essentially Eₖᵢₙ-Eₚₒₜ) to start a theory; 2) Starting from S, one can demonstrate that energy conservation follows from time translation invariance w/o explicitly resorting to forces and the "force times distance" definition of work.

> If I understand you correctly, you're saying that the little proof in this video demonstrates that the laws of mechanics in question are time-independent ("invariant under time translation"). Is that right?

Sorry, I should have been clearer on that. Firstly (as a minor point), physicists usually use the label "time-dependence" for quantities like forces and energies, and the term "time-translation invariance" for laws. Secondly (to answer your question), the video's premise of the proof is that the **forces in question are not explicitly time dependent** (0:27). The more general concept of time translation invariance is only touched at the very beginning and the very end (3:47) of the video.

> If so, what would they look like if they were time dependent. ... Is there an easy way to demonstrate how a single force could cause KE to increase at a rate different from the corresponding decrease in PE if they weren't invariant under time translation?

If the force is explicitly time dependent (contrary to the premise in the video), then the potential energy could indeed change at a rate differently from the kinetic energy. The key argument is alluded to in (1:40) "You can't have a potential energy for a force that changes over time". The potential energy can indeed change due to factors that have nothing to do with the body's kinetic energy but e.g. due to temporal changes in the strength of the force field (gravitational, electrical, etc.).

For example, consider a large conducting sphere with a positive electric charge, and a probe with small negative charge that moves away from the sphere. The probe has an initial kinetic energy, and as its distance x from the sphere's center increases, the potential energy due to the electric field increases by ΔEₚₒₜ while the kinetic energy decreases by ΔEₖᵢₙ (same is true for a mass moving in a gravitational field, but I use the electrical field for a certain reason that becomes clear below).

If the force F in the electric field is **not explicitly** time dependent, then the magnitude of F decreases by ∝ 1/x² (like a mass in a gravitational field), and the force is only **implicitly** time dependent via the time dependence of x=x(t), meaning: **F=F(x(t))**. In this case, **ΔEₜₒₜ=ΔEₖᵢₙ+ΔEₚₒₜ=0** (as shown in the video).

If the force F is **explicitly** time dependent, then the force can change over time also if x remains constant: **F=F(x(t),t)**. For example, imagine the electric charge of the sphere is leaking away as the probe moves. While the probe's distance increases, the probe uses kinetic energy ΔEₖᵢₙ to do work against the electric field as long as there is some charge left on the sphere. But once the sphere is completely discharged, there is no force left, and the final value for Eₚₒₜ is independent of ΔEₖᵢₙ, meaning: **ΔEₜₒₜ=ΔEₖᵢₙ+ΔEₚₒₜ≠0**.

Now you might say: There might be a better way to account for the total energy, say, by including subtle effects coming along with the discharge of the sphere. I need to think about it, but that's beyond the point for this demonstration. Plus, there are other ways to introduce time-dependence: imagine that for some freak reason the charge on the sphere suddenly disappears (thus also violating charge conservation), or that the number of spatial dimensions n suddenly change (F∝ 1/x² is true for 3 spatial dimensions and becomes 1/xⁿ⁻¹ for n spatial dimensions). Those are special examples in which time dependence of the force is induced by violation of symmetries of certain natural laws. Whatever you chose, it should now be clear to you that ΔEₖᵢₙ+ΔEₚₒₜ is not conserved when the force is explicitly time dependent.

So, while I'm at it, let me also show that the video's conclusion (2:23) ΔEₜₒₜ/Δt=Fv-Fv is only true for time-independent forces, but for this I need to use calculus (you can skip the proof, if you want, but at the end you will see that the result is Fv-Fv+"something").

The rate of change of the kinetic energy is

dEₖᵢₙ/dt= d(½mv²)/dt= ½m2vdv/dt = vma = vF(x,t) = vF,

as derived in the video w/o calculus, and doesn't depend on whether or not F is explicitly time dependent (we've just used Newton's law).

The rate of change of the potential energy if the force is not only explicitly x dependent (∂x) **but also explicitly time dependent** (∂t) has then two contributions:

dEₚₒₜ/dt = [∂Eₚₒₜ/∂x dx]/dt + [∂Eₚₒₜ/∂t dt]/dt

(dx/dt=v, dt/dt=1)

= ∂Eₚₒₜ/∂x v + ∂Eₚₒₜ/∂t 1

= ∂/∂x (-∫F(x,t)dx) v + ∂/∂t (-∫F(x,t)dx)

The first term is the derivative w.r.t. x of the familiar expression for work and formally reduces simply back to the force F(x,t); the second term is the partial derivative of the potential energy and remains a time-dependent function:

dEₚₒₜ/dt = -F(x,t)v - ∂/∂t ∫F(x,t) dx.

The rate of change of the total energy is then:

dEₜₒₜ/dt = dEₖᵢₙ/dt+dEₚₒₜ/dt

= vF-vF-∂/∂t ∫F(x,t)dx

= - ∂/∂t ∫F(x,t)dx

and **is only zero (i.e. energy is conserved) if ∂F(x,t)/∂t=0 (i.e. if the force is not explicitly time dependent)!** Only in this case we obtain dEₜₒₜ/dt=Fv-Fv, as demonstrated in the video.

But remember that, as I mentioned at the very beginning, there is a more general way to tie time translation invariance to energy conservation without resorting to time independent forces in the fashion the proof in the video does.

> I guess the thing that is confusing me is that my impression of those equations is that they are assumed to be constant. ... How is the assumption of invariance under time translation not baked into the equations?

They are indeed "assumed" to be constant in general but not because writing A=B+C magically makes the sum B+C constant by definition but because 1) of empirical evidence that verifies B+C is a meaningful bookkeeping constant under certain conditions (see above), and 2) there are deeper theoretical arguments (time translation invariance) that justifies this assumption. If B is the kinetic energy and C the potential energy, then laws that are not time translation invariant could e.g. make B changing willy-nllly in the absence of any force field (i.e. C=const), thus not conserving A.

@Diiffer > Aren't the laws of mechanics invariant under time translation by their formulations in the first place?

Yes, for forces F that are not explicitly time dependent (∂F/∂t=0).

> If, for example Newton's second law WAS time dependent and acceleration was inverse proportional to the age of the universe, wouldn't the time dependence have to be stated? Wouldn't Newton's second law had been something like this: (sigma) F = k*t*m*a were k is a constant and t is the age of the universe.

This would be indeed an example of an explicit time dependence, ∂/∂t ΣF = Σ ∂/∂t tkma = Σkma ≠ 0. But there are also more examples. Consider a ball thrown upward against the Earth's gravitational force, which is approximately F=mg if the ball remains close to the surface. Then

dEₜₒₜ/dt=dEₖᵢₙ/dt+dEₚₒₜ/dt

= d/dt(½mv²)+d/dt(mgx)

= -½m2vdv/dt+mgdx/dt [v=v₀-gt if F is chosen positive]

= -mgv+mgv=0

But this is not true if, for some freak reason, Earth's gravity on the surface starts to change at a time t=t₀. Let's say it was constant for t<t₀, g(t)=g₀, and from t>t₀ it suddenly decreases, say, g(t)=g₀/(1-k(t-t₀)) (k being a constant). Then gravity gently disappears, g(t→∞)=0, and in the process of it,

dEₜₒₜ/dt =dEₖᵢₙ/dt+dEₚₒₜ/dt

= d/dt(½mv²)+d/dt(mgx)

= -mg(t)v + mg(t)dx/dt + mx dg(t)/dt

= 0 + mxg₀ d/dt(1+k(t-t₀))

= -mxg₀k/(1+k(t-t₀))² < 0 [for all t>t₀]

Meaning: With increasing time t, more and more of the total energy disappears (first faster, then slower).

If you don't like the math, just imagine a simplified example where you throw a ball upwards in a constant gravitational field, watching how kinetic energy is consumed to do work against it, thus converted into potential energy. As the ball reaches its turning point and all the kinetic energy is converted to potential energy, God comes and is so mean and abruptly turns off Earth's gravity. The total energy is now zero (or any other arbitrary constant unrelated to the ball's initial kinetic energy). This is another example for a time-dependent force not conserving energy.

> My problem with this video is that it tries to prove that a quantity - which comes in two forms, differing only by a minus sign - is conserved by adding the difference over time of each of the forms together. Of course they will equal zero, since something added together with minus one times itself is nothing.

As I explained to @iamihop, the correct expression for explicitly time dependent forces (∂F/∂t≠0) is

dEₜₒₜ/dt = Fv-Fv-∂/∂t∫F(x,t) dx

(in our above example, when t>t₀)

= -∂/∂t ∫ ma dx = ∂/∂t ( mg(t)(x+const) ) [a(t)=-g(t)]

= mxg₀ ∂/∂t 1/(1+k(t-t₀)) [partial derivative]

= -mxg₀ k/(1+k(t-t₀))²<0

(result like before).

> (You could also "prove" that the total velocity of an object is conserved by defining a form of velocity called phantom velocity, which equals minus "normal" velocity. Add the difference in "normal" velocity over time with the difference in phantom velocity over time and you get zero. The total velocity is the same! :D Q.E.D!)

If you multiply velocity with mass, you have indeed a conserved quantity called momentum: p=mv: Σᵢ pᵢ = const. From Newton's laws you can then directly show that the velocity of an object with constant mass is indeed conserved when no force is acting on it (F=dp/dt=ma) without resorting to phantom velocity but actual velocities. The difference to energy conservation is that all momenta "look the same" (making momentum conservation trivial), but energies appear different. Potential energy is bound to a potential while kinetic energy is not.

> We need to know that kinetic and potential energy are two forms of the same quantity

I would say "of the same quality". Energy conservation requires more than momentum conservation does.

>, a quantity that actually exists and enables an object to do work.

Heat is also energy, but only part of it is usable to do work (second law of thermodynamics).

> It is essential to a proof of conservation of energy to first prove that the work an object can do by reducing its speed to zero (i.e kinetic energy) is 1/2 m v² (or simply the work done on the object when taking its speed from zero to v).

It seems you are afraid that the definition of work W=∫Fdx creates circular logic. Note that work is, by definition, tied to a force F acting on an object, thus changing its position x.

W=∫Fdx = ∫dp/dt dx = ∫mv dx/dt = ∫mvdv= ½mv² + c

> One must also prove that the work an object can do by moving in a force field with the force (i.e potential energy) is -W (the work done on the object when moved it against the force).

Same definition:

W=∫Fdx = ∫-mgdx = -(mgx+c) (if g is constant and force is anti-parallel to movement)

> All the proofs I had seen of these formulas assume conservation of energy,

I hope it is now clearer that the video and the definition of the energies do not make this assumption. Of course, energies are, like momenta and other quantities, invented to express constants of motion.

> until recently. I found out that you can prove the formulas using Newton's third law and the definition of definite integrals. When doing so, it wasn't hard to see that the work an object can do by reducing its speed to zero is the same work done on the object when taking its speed from zero to v.

Exactly.

> It was just as easy to see that the work an object can do by moving in a force field with the force is the same (but negative) work done on the object when moving it against the force.

Right.

> I have proved myself wrong, Henry's proof is not circular.

But don't forget that the proof only works if forces are not explicitly time dependent, as I tried to show above.

> But sadly he doesn't explain why it isn't in the video himself.

The video is not a lecture. It just wants to give people an idea about how conservation is tied to time symmetries. The relationship between symmetries and conserved quantities is a really big deal in modern physics. It's part of the foundation of particle physics theories, and more.

@NeedsEvidence

Thank you for replying. I see that the proof only works if forces are not explicitly time dependent, and supposedly you don't need to account for forces that ARE time dependent, because potential energy doesn't work for these forces. Why? Where does the potential energy of a permanent magnet in a magnetic field caused by an electromagnet go when the electromagnet is turned off? Where does is come from when it is turned on? It would seem you could make a perpetual motion machine: Let the permanent magnet fall in the field, turn the electromagnet and the field off, take the kinetic energy of the permanent magnet and store it somewhere else, move the permanent magnet back to its original position without fighting a force, turn the electromagnet on and repeat the process. Do you know what is wrong with this?

It is lost in an object but the energy is universe remains constant

basically used maths to prove maths..

No, its a law in physics....

Does "no absolute time" imply "time is not one-way"?

Or, rather, WOULDN'T "no absolute time" imply time is not one-way?

Finally some math.

Cool video, but there's an important assumption made here. Namely that the only forces applied are those associated with the potential energies involved. Otherwise the F from the kinetic energy is net force, butd the F from the potential energy is not the net force. The energy of a system can always change, if there is external work done (forces other than those associated with the PE).

In fact this is tautological since we define KE from the idea of the net work done and the work energy principle, and if you define the change in PE as negative the work done by the "net force" then you have simply assumed energy conservation by definition. Basically we got so mixed up with words and language, we thought we said something profound, but really just restated what we already assumed.

Noether's theorm..

As in No way I'll do a Laplace transform to prove that.

I know I'm 4 years late, but that's Lagrangian.

this is so satisfying

"i dont get it which means its stupid."

-professional youtube comment section critic

"It's not like there's some absolute starting time."

What about the start of the universe, that sounds pretty absolute to me?

Understood this the second time I watched

"Simple" lol idk I love these videos when I hardly understand most of them. It makes me feel smart and dumb at the same time.

"Simple" I guess I understood most of it and I'm in 8 grade my class haven't even done this stuff yet... I'm proud of myself...

Thank you, Henry! :)

TIL that when I'm drinking wine, isn't the time to watch MinutePhysics.

**"SIMPLE"**

So cool! Takes my knowledge of physics to the next level, that's awesome!! :D

When I compare my energy in the afternoon to my energy at night, i should see no difference, because time does not affect energy, brilliant!

Not quite. The energy of a closed system doesn't change. You aren't quite a closed system. If you were to say eat an apple you'd gain the energy of the Apple. :)

This is literally THE most confusing video I have ever watched :(

That post-3:20 calculation was quite an exciting glimpse into conservation of energy.

U shud hav made it part of da original video

Thank you very much, MinutePhysics! I love this channel very much! You are doing a great job, and these illustrations are awesome!

Great job with these mechanics videos! I'd love to see a video tackling the Twin Paradox!

"Simple" he said

I like to assure myself that I understand what henry's saying.

Wow !

This has been expressed in a very beautiful way using mathematics !

nice try pronouncing the "Ö" :P

Finally, the kind of content I've expected out of minutephysics for a long time

This was personally one of my favorite videos in a while. Physics is mostly math, and it's nice to see you show some skin every once in a while (metaphorically, that is).

Great job Henry!

"If you can't explain it simply, you don't understand it enough."

~ Albert Einstein

It's been such a long time sense you've last made my brain sad. Well done.

A bit too fast maybe, even when rewatching, its hard to keep up

This is really awesome. This makes me wish I was a Physics major.

That's really interesting, I wrote a collision engine in my game that works on the same "At any particular time the force must remain the same" idea.

I have an exam of classical mechanics next Friday, so hello Lagrangian :')

ya ....

tats “simple” proof of conservation of energy ......

“simple” =_=

although i have no idea about all those concepts, i still thank you for sharing this to us :D

This is brilliant! It all makes sense now! I love to live!

Thank you, now i get what my physics teacher tried to tell us in class. You are awesome, keep up the good work.

Before I hit the replay button 17 times, could you make a video on potential energy? That whole thing doesn't make sense to me.

Think of potential energy very literally. It's energy that has the *potential* to be kinetic, i.e moving.

Hi, potential energy sounds confusing but you see examples of it in your every day life. The best visual example of potential energy is a spring. When you stretch a spring it takes work. The spring fights against you, it does not want to be stretched out. When you let go of the spring it snaps back to normal. What you are doing when you stretch the spring is pumping energy into it. when you let go the spring uses that energy to bring itself back to normal unstretched condition. Kinetic energy is the energy of motion and potential energy is stored energy. When you jump into the air you use energy to gain hight. But as you go up you slow down and then come to a stop. Where did your energy go? When you started your jump you were moving up verry quickly but now your not moving at all. All that energy went into gravitational potential energy. All that meens is that gravity took your energy in your upward motion after you jumped into the air and stored it (temporarily). You only are motionles in the air for just a moment, then you fall back down. When you fall down you gain downward energy of motion. But where did gravity get the energy to make you move quickly in the downward direction? Gravity uses the potential energy it robbed from you initialy. Just to repeat: when you jump you have energy in the form of upward motion. Gravity robs you of this motion bringing you to a standstill in mid air and stores the energy temprarily. Then gravity uses the stored energy to send you quickly back down to the ground.

@Isaac Thefallenapple Isaac, the fallen apple has less potential energy. lol

@Giles Bathgate Hehe, never thought of that =)

Thank you all for explaining this to me while being nice. I think I got it now.

DonkeyxKong93 - 2014-12-22

A physicist sees a young man about to jump off the empire state building and he yells, "Don't do it! You have so much potential!"

Sorry I had to tell that joke.

Boris Esquilo/tromp - 2018-03-31

I know this is fuckkng old but this should have a lot more likes 😂

Mehul Jain - 2019-09-06

Ik it is 4 years old but it is still very funny 😂😂😂