Calculus Proof that a=v^2/r

[soleilenvierge]

I don't understand how is can make a derivative of a function (let's say of: r*cos(teta) ) in regards to a variable (t) that's not even part of the function ? I tried in Matlab and it won't let me. In matlab, you have to make a derivative in regards to one of the functions variable. Can someone explain?

[baboyako1000]

is this for uniform circular motion only?..In your proof it wasn't assumed that the magnitude of the velocity is constant because it wasn't needed, so it must apply for non-uniform motion as well,so i'm wondering why the magnitude of the acceleration vector exactly equals v^2/r..shouldn't that be the magnitude of the centripetal acceleration only... so that the magnitude of the overall acceleration should be different??im really confused .....can someone please reply....thanks in advance!!

[avatar098]

thank you(:

[Clipster15]

Thank you. My Physics teacher did this a couple times in class but it went all to fast for me to get a good understanding. I was going to try and prove it myself, but I knew you probably had an intuitive way of showing it. Thanks!

[johnqwertyful]

w or omega?

[anonymous345]

@johnqwertyful

omega

[andyscape]

well i just graduated high school, and now after this video i have decided to sell meth for a living.

[iseefurtherthanyou]

@andyscape lol

[misterman567]

great video!

p.s. omega! not w!

[46dvmehta]

I love you man! You are a real GURU!!!

[cooldude8081]

Which program is this where u can draw those math pictures?



[trollwarlord3]

you fucked my mind!!

[bmoua007]

i just wanted to let you know that it took 3 pages for my professor to prove this, and you did it in like 8 lines. thank you so much!

[powerseriesman]

this is awesome, i love this!! why do we need to go to these annoying schools if we can get our education online. the only thing i guess is chemistry experiments, but we can buy chemistry sets and do them at home, or maybe rent space and do the experiments there (without an annoying lab partner)

[MarvelsofaLifetime]

That's what I call a proof. It's always great to watch these and then learn how to prove them for yourself; because that way you'll never forget them.

[herojha]

that's what we set out to prove and now i am out of time, lol!

[levistamp]

wow great vid. great proof.

[levistamp]

wow great vid amazing proof

[DawsonMan21]

really great video. Kills my professors, who i'm paying thousands of dollars a year to teach me this.

[pass135]

You sir are a legend of man, i'm officially your newest student! brilliant vids, can't thank you enough for how this helps :)

[nuuts88]

nice explanation man. congrats. 

[liu408]

hey, shouldn't that be "omega" instead of "double-u"???

[sharpaim23]

I think my brain exploded while listening/watching the video..

[antonyneal]

Nice i was curiouse because i knew there was a proof and it felt like it was on the tip of my tounge but it wouldent come out. Any ways bravo.

[fas890]

I got lost at the end but I figured it out since I skipped a few videos. dθ/dt is the change in θ over time. In other words, dθ/dt is the angular velocity, the rate at which the angle changes over time, or v/t. This is represented by w. so w=v/t. where v is the velocity vector. how awesome.

[rinwhr]

For a minute there, I don't know why I went into Polar coordinates and I got confused lol

[n1a1s1i1m]

I smell perfection. Spot on, mate!

[killroy74]

I subbed. Im going to be in calc next year and your a very good teacher IMO.

[onlyAerik]

Eh, despite how damn good it is, it might not be a good idea to sub. You'll receive videos you're not ready to watch yet. You'll be going backwards to watch what you need for class anyways.

[killroy74]

yeah i know i wont watch them until i need them. i just want to make sure I remember they are available here.

[patilnikh]

Sal, you have the angular velocity listed the next of this, so how do we know angular velocity first?

[im4everskilled]

It all makes sense and I totally agree.

(I have no fucking clue what the fuck you're talking about)

[stanthaman09]

LMAO!! tht was funny dude, but makes sense to me :)

[Yu2Kal]

How can you not get it, this calculus is very well done and easy to follow. I understand some people don't understand calculus thats why he has done a visual proof of the same thing so you have a choice of what you prefer. Atleast have some respect for this guy, he does a hell of alot of videos for people for FREE. This guy is awesome! He makes it so clear, this guy is better then my old tutor and I done have to pay for this. So thank you Sal for these videos, I appreciate it alot!

[rctriplefresh5]

he lost me at 2 minutes in

[saxmaster2]

I just wanted to say thank you. I'm a Math/Computer Science double major and I won't accept any formula anymore until I see the proof of it through calculus. Thanks to you, I now have. It's rather sad that this video only has around 14500 views though...I would have thought that there would be at least a 50000.

[madscirat]

Sal you are my hero. Thanks to your calculus videos I understood all of this perfectly.

[MajorUtah]

I forgot to remind you, that if someone tells you that you look taller today, you won't wonder why.. ... (*Xy'

[michaelcweir]

I have posted a 1 page proof of Fermat's Last theorem for you all to view. Either view video above or click on my name michaelcweir to view

[gqstan1]

a great man did this

[hgeithus]

This was actually pretty neat :)

[AaronLIEF]

you really know your stuff Sal.

I can really learn alot from you than my other teachers

[osiris392]

thats funny i keep reading the same thing. our teachers suck and the online random guy knows how to teach better than them. i love Florida and all its poor glory. we're dead last in the U.S. states for the least amount of money alotted to education.

[xlwaiyiplx]

wat? florida still exist? unbelievable!

[maujo2009]

There's a very beautiful proof for the equation of the centripetal equation which does not employ calculus. It's very intuitive and based more in Physics principles and provides a nice interpretation of this concept as "the acceleration of falling around instead falling towards", just like our moon falls around the Earth, but not towards it.

[bottle2lip]

humbled

[buzbyX]

that's awesome

[khanacademy]

|v|/r is the same thing at dtheta/dt:

d(arclength) = dtheta *r (definition of a radian)

|v| = d(arclength)/dt = dtheta*r/dt

|v|/r = (dtheta*r)/(dt*r) = dtheta/dt

[gangs30]

haha entrez got served

[stan021]

very nice video.

[punkzaznboi]

cool

[Magnesimproduction]

that was way over my head

[bean2594]

At 4:05 you should have an equals sign between your time derivative and bracket notation

[shurakbh]

Showing this using vector components etc is clunky, ugly, and masks the physical significance. Chain rule. Yeah.

The change in velocity is delta_V=2Vsin(theta/2), the time in which that happens is delta_t=theta*R/V (given theta in radians and constant V).

Thus delta_V/delta_t = (2V^2/R)sin(theta/2) (*), which in the limit delta_t->0 replaces (*) by dV/dt = V^2/R.

[shurakbh]

Oops, seems like you already did that in another video. My bad. Still, no excuse for this atrocity. :)

[MedwayMan1945]

This can be done easily using complex numbers.

Put z = re^(i theta) and differentiate it twice with respect to t.

z dot = r i omega e^(i theta) = i omega z

z dot dot = (i omega)^2 z = -omega^2 z

[FatherAbyss]

That is so elegant, thanks for sharing. I myself appreciate to know as many ways as possible to mathematically show the exact same physical principle. Great stuff.

As another note, you could also define a unit vector e as cos(theta)*x+sin(theta)*y, where x and y are unit vectors.

Then you start out with the function:

p = r*e, then

v = r*omega*e^, where e^ is the perpendicular-vector to e (because e^=(-y,x) where y and x are the coordinates of e), and finally:

a = r*omega^2*(-e) = -r*omega^2*e.

[FatherAbyss]

That was meant as a reply to

MedwayMan1945's comment on using a vector in the complex plane.

[tharock220]

Show me an application for gradients.

[pookellypoo]

Wow, that was neat.

Didn't know you could take the derivative of a set of coordinates like that :o

Thanks Khan :)

[dorcha2442]

SRRY......i mean VALUE!...............not vale.

[dorcha2442]

When you said the position "p", that reminded me of this funny thing my math teacher did once. To better help us understand the value of "p", he made some jokes. The first thing he said was "What do we have to do to "p" to find out the vale?" Get it?

[Ibeatyou23]

this is a2 physics for the person who asked.. and it is a biotch

[footlover312]

oooouuuuuucccchhhh

[Blackdog1015324]

my head hurts

[SteadyHaze]

What level of physics is this?

[magipablo]

we've done this in A2 physics, although it wasnt proved at the time

[wotchadave]

Yeah I would say it would be possible at A level to figure out although taught at University (Perhaps?)

Incase you're American, A-level = College

(I know how the education systems confuse me o_O)

[eldominicanboy]

thanks man

[Tymkew1]

Not a bad proof lad... however I should make you aware of a slight notational discrepancy. Angular Velocity is usually denoted by the Greek letter Omega, and it only bears the likeness of a "W." Having settled that slight pet peeve of mine, I admit I thoroughly enjoyed your video. Good job Lad!

[stanislovely]

I like how a nineteen year old, who couldn't be past second year physics is correcting someone who is clearly at least in their early 30's and has far more extensive knowledge of the topic....

[Tymkew1]

tsk tsk... I admire the man's work, and never intended to "correct" him. By convention Angular Velocity is denoted by an Omega, and that is all I was trying to say. This example is abstract anyway, so it doesn't matter what symbol is used. Having said that, I clearly must have offended you, since you are so sensitive to criticism that wasn't delegated to you to begin with, so I apologize.

[stanislovely]

I was just giving you crap man. But your comment did come off as rather arrogant. Maybe that's an easy thing to do on the internet; my girlfriend tells me I sound condescending all the time. *shrugs*

Anyway, I'm guessing the reason he didn't use omega is because of how annoying it is to key in. We've got Unicode, and I can read my .pdfs in Japanese, but for some reason we still don't have user defined keyboards!

Computers = frustration. They always overlook the simple things.

[petergeras]

w is not only the english letter we know, but ALSO the greek letter omega. It would be the small-case of omega, while the capital is the one you're probably more familiar with.

[Tymkew1]

...I'm not sure I understand what you are trying to say. I am well aware of the shape of both cases of Omega, and it being Greek, is not the same as the English letter "w".

[mkhalidaziz]

Thanks from Pakistan! You are a genius!

[spider853]

hi.. in this formula Ac = V2/R .. the Velocity is a scalar? if its scalar then Ac will be scalar to. So if we want to convert Ac to vector we need to make something like [-Ac * cos(Theta),-Ac * sin(Theta)] ?

[xzxz619]

Why does he say, "With the respect of time", please fill me in.

[hutzelmann]

because the angle changes while time flys by...

[SycophanticLeech]

It's actually "with respect to time"

[SerjicalProdigy]

velocity is a vector...

[payasitopimpim2002]

amazing video my friend....thanks from lima peru

[Waranle]

Thank you Khan!

[idricool]

I learned a lot too ! Thanks !!!!

[Oxydox]

thankyou khana! This is so perfect for what I'm learning right now :D