Pi and Bouncing Balls - Numberphile

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Our Pi Playlist (more videos): bit.ly/PiPlaylist
Professor Ed Copeland on a strange occurrence of Pi involving bouncing balls.
More links & stuff in full description below ↓↓↓
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OUR PI VIDEOS ---
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• Pi - Numberphile
BOUNCING BALLS AND A STRANGE PROPERTY
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• Pi and Buffon's Matche...
THE SOUND OF PI
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7 сен 2024

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Комментарии : 1,2 тыс.

@numberphile 12 лет назад

A computer simulation of this experiment has been posted at our "extras" channel nottinghamscience and I'll put a link in the video description

@numberphile 10 лет назад

Ed has added this for people who want more detail: www.numberphile.com/pi/Pi_Balls_Ed.pdf

@alexandrumoise1511 9 лет назад

That's a beautiful demonstration. Very nicely put together, fun maths indeed.

@terrymoore4238 9 лет назад

+Alexandru Moise It's a veritable tour de force. That's its problem. The momentum and energy equations can be solved in four lines--it shouldn't take four pages. (The crucial idea is to factor the change in energy using the difference between squares, then note that there is a common factor given by the momentum equation. This then gives two linear equations for which the solution can be written down at sight.) And, in this case, the power of the transformation doesn't need eigenvalues (although that is a useful technique for many problems). Instead, by scaling one of the velocities, the transformation is seen to be a rotation. Combining a number of rotations through the same angle is trivial. I hesitate to criticise someone probably much brighter. Ed's proof is the right approach--the problem lies in the unnecessary detail.

@alexandrumoise1511 9 лет назад

Terry Moore well I noticed that it's, as you put it, a tour de force, but I liked it and I see how it can be applied broadly for many other problems, besides, on top of not making shortcuts in the maths, he explains every step using words so that's how it takes up so much space.

@alexandrumoise1511 9 лет назад

I think

@terrymoore4238 9 лет назад

Alexandru Moise Yes, it's all explained in words. But it's more complicated than necessary. Solving the equations by guessing that the solution is linear and solving for the coefficients is much more involved than the direct approach. Using the difference between squares, we can reduce the problem to a pair of linear equations. The next simplification is to rescale so that points move around a circle, from which the result of n collisions is immediate. The rest of the proof is the same as Ed's.

@LakeNipissing 8 лет назад

Professor Ed just comes across as the nicest bloke.

@gigamungus5764 5 лет назад

3blue1brown's video sent me

@rikadomez8201 4 года назад

Sorry for being late

@chriskohchannel 4 года назад

yea

@filgiupo4853 4 года назад

yeah me too

@I-am-duck Год назад

same

@Lani-5 Месяц назад

me too!!!

@numberphile 12 лет назад

@MrStrangeHumor me neither... but it's still interesting isn't it...?

@mattfreeman927 8 лет назад

Love this man. I've just read his full resolution to this problem and it's astounding! *I know that you'll probably never read this coment, but I just wanna say that you're AWESOME!!!!!*

@kenhaley4 6 лет назад

Absolutely amazing! I verified this with a short Python program, running N up to 9, giving a result of 314,159,265 collisions. Exactly matching the first 9 digits of pi! The program runs in about 5 minutes. For N=10 the result would be 50 minutes.. I didn't run it that long :-)

@SongADayPodcast 10 лет назад

I love these amazing kinds of 'coincidence' in math. It's like a treasure hunt - digging deep into math and finding amazing gems sprinkled about.

@graspee 10 лет назад

I wouldn't draw on that absorbant looking paper with a pen of that type on that table- I'd be really scared it would go through and mark the felt.

@datenegassie 10 лет назад

Half-Life Pi confirmed.

@donaldasayers 6 лет назад

Where did the cos(theta) come from?

@bowtangey6830 2 года назад

Yes, and in the context of our problem what IS theta?

@numberphile 12 лет назад

@HappyBeanBag yes

@seav80 5 лет назад

I just came here from the 3blue1brown video on the same topic. What mystifies me is why this video uses 16×100ⁿ×m while 3b1b omits the 16.

@fadinghalo2387 5 лет назад

seav80 The only difference I noticed is that they use different shapes, but I guess it has nothing to do with momentum and kinetic energy.

@ConnorDuzMinecraft 5 лет назад

"It has a factor of 16 throughout which shouldn't be there, unless they are making some added assumption I'm unaware of." That's what 3b1b's description says. Maybe he will clarify it in his proof video or I can ask on r/3blue1brown later on if he figured out why.

@jackpisso1761 5 лет назад

They're not counting the same thing. 3b1b is counting the total number of collisions, while here they're counting until the large ball changes direction.

@ShivaramakrishnaReddy 5 лет назад

There is no mystery. 3b1b is counting the TOTAL number of collisions, here they are counting the Required Number of collisions for the big ball to reverse it's velocity

@frxstrem 5 лет назад

They only count one fourth of the bounces in this video: they don't count the bounces after the big ball has stopped, and they also don't count the collisions between the smaller ball and the wall. So you need four times as many bounces to get the same number, and since the number of bounces is proportional to the square root of the ratio of the masses, the ratio of the masses needs to be 16 times larger to get the same number as in 3b1b's video. It's essentially the same problem/proof, but they're just expressing it in different ways so you get a factor 16 difference in the masses.

@user-cz7bu5qk8w 11 лет назад

In this problem, cos(theta) = (1-m/M) / (1+m/M), so if m/M = 1/16, then cos(theta) = cos(0.88235 radians), or cos(50.5551 degrees). What goes into the final equation is cos(n*0.88235). The cosine function doesn't have to be used to relate angles. It can be used for any situation in which some variable follows a cosine/sine wave pattern, regardless of angles being involved.

@cwjakesteel 11 лет назад

Wow this is amazing. I thought I would never see pi again outside angles! But technically it is angles because of the cos function.

@numberphile 12 лет назад

@qotsaandsoadfan1 google periodicvideos and brown paper

@elevown 10 лет назад

Obviously they dont demonstrate with the real balls- its a maths thing not physics- it assumes 100% elastic collisions and conservation of momentum- neither of which are possible in real life.

@punya1621 6 лет назад

Also friction and size. Btw, the table would break with that heavy a ball!

@grantsdaman01 6 лет назад

It is still a physics thing if you can create a physical analogy of the math. Mathematical principles are what determine this, not ease of demonstration through physical elimination of variables. If you've ever taken physics in school, you'd know that creating "unrealistic" scenarios such as this is what comprises most of the study.

@anandsuralkar2947 5 лет назад

That doesnt mean its not possible physically its very very possible

@ishanr8697 5 лет назад

Conservation of momentum is definitely possible in real life! However, in this case some momentum can be transferred through the table to the Earth.

@AhnafAbdullah 5 лет назад

Perfectly elastic collisions do exist, for example in colliding gas molecules

@numberphile 12 лет назад

@FHomeBrew what do you think?

@hindurashtra63 10 лет назад

This is with all these Assumptions & Conditions : - Perfectly smooth surfaces on Both Balls - Ignoring Surface Resistance - Ignoring Air Pressure - Ignoring Reflective co-efficient of the Reflective Surface - Assuming a Pre-defined relation between little ball and small ball In short.. All these would never happen in real life, Only in a perfectly controlled experiment in a vacuum chamber !

@numberphile 12 лет назад

@MA2ANDA our super friction-free collision machine was broken that day! :)

@Swaza13 12 лет назад

Wow! It's amazing how everything relates to each other in maths and physics. :)

@ImaginePeacePro 11 лет назад

Every time I watch a Numberphile video I say "Oh I'll just watch one more." And then the next thing I know it's 1 in the morning and I'm debating someone on what the coolest prime number is or if zero is even. Gotta love Numberphile.

@SocietyIsCollapsing 7 лет назад

Great video. Also, at 6:11, there's an error on the paper. the 5th N should have 2 more zeroes, and each one thereafter.

@shawnpheneghan 7 лет назад

In the elastic collision the little ball loses no kinetic energy - but it definitely changes momentum. Momentum is a vector quantity and and the difference before and after the elastic collision is 2x its original momentum

@christianlawrence2714 5 лет назад

Anyone else here looking for references for the latest 3blue1brown vid?

@shauryanrana6165 5 лет назад

Exactly but the two are a bit different...

@Bibibosh 5 лет назад

Christian Lawrence me

@JWY 12 лет назад

When you graph the speed of one ball against the other, either way, you get a smooth quarter circle (squashed, as the balls speeds range over different extents). The top of the circle is where one ball is stopped or stopping and the other ball is fastest. There's exactly the same angle (as unsquashed circles I mean) between each point. The mass ratio of 16*100^n is like 4^2 * 10^2n, and I guess the 4 fits the quarter circle, the 10^n brings digits up into bounce counts, and squared is physics.

@atharvas4399 5 лет назад

here from 3b1b?

@GameNOWRoom 5 лет назад

Yes

@cinephilia_3099 5 лет назад

Yup

@sumanthvasista4998 2 года назад

For those interested, it is possible to derive this from first principles of conservation of momentum and energy as well (Had a fun day figuring that out) m = 1 u = 1 v = 0 n = 4 M = 16*100**n count = 0 while u > 0.0001: v = v + 2*M*m*u/(m*M + m**2) u = u - 2*m*v/M count += 1 print("Number of times collision occurs = ", count) print("Therefore pi ~", (count+1)/(10**(n)))

@Sarnetsky 10 лет назад

4:42 - 8:36 can be completely omitted with no loss of thought whatsoever :)

@pierrestober3423 10 лет назад

that is the only problem with numberphile, sometimes they just overexplain what is already obvious

@Sarnetsky 10 лет назад

Pierre Stöber true, but there's the other side of this problem: there are people who don't quite get things even if they are overexplained :)

@StefanReich 6 лет назад

Yes

@blacktimhoward4322 4 года назад

This comment can be ignored with no loss of thought whatsoever

@Blackary1 11 лет назад

I would love to see a video or a link with a derivation of the equation for the velocity after N collisions

@derekcapeles4911 9 лет назад

I would love to see this as a computer simulation.

@Emil-yd1ge 9 лет назад

me too !

@churchmanner 9 лет назад

Yeah me too. obviously the later ones would be sped up (nobody wants to watch 3,141 collisions at normal speed)

@adamakbar4277 9 лет назад

Yeah that would be an amazing idea.

@lorebreaker4970 7 лет назад

Great idea... we should fund this!

@tj1990 6 лет назад

i think you mean an animation.

@wowsa0 12 лет назад

@PacificCircle1 If you want to see a circle somewhere in this situation I think it helps to think about what happens if you plot the speed of the first particle against the speed of the second. When you apply energy conservation you'll get that the system at any time has to lie on an ellipse in this diagram. Rescaling the axes can get you a circle if you want. The system is hopping round this circle every time there's a collision and I think that's a good way of seeing where the Pi comes from.

@OnionKat 9 лет назад

I kinda want to be a mathematician.

@Tyngdlyftning1 9 лет назад

Why arent you one then?

@Tyngdlyftning1 9 лет назад

***** Engineering is better to study. You get better jobs and you can easily study some extra physics later.

@MrLC92 9 лет назад

***** Oh, don't worry, I can live with that.

@SkyFoxTale 9 лет назад

ᅚ ᅚ Better jobs does not constitute "better."

@zTheBigFishz 8 лет назад

+SelfAudioBook Slight correction: go to the university and study math!

@johnodonnell4593 10 месяцев назад

Error at 1:43. It’s not just “count until the big ball starts going the other way”. It should be “keep counting until there are no more collisions” which will only happen when the big ball velocity becomes greater than the small ball velocity.

@DeathBringer769 6 лет назад

Was that a song made using the digits of Pi at the end? Sounded like notes assigned to the infinite values of randomness of Pi somehow ;)

@numberphile 12 лет назад

@WhiteHenny the written bits are the fault of the very overworked and non-mathematical film-maker... he accepts all criticism with humility and embarrassment!

@plasmacrab_7473 8 лет назад

Pi + 10 is the length of the video

@otesunki 7 лет назад

Pi + 10.01

@plasmacrab_7473 7 лет назад

Pro Odermonicon Really? For me, it's 13:14. It's RU-vid's approximations, I guess.

@nerdycubing2934 7 лет назад

for me it's 13.14

@valeweinmann9907 6 лет назад

PlasmaCrab _ OMG that's rite!

@LudwigvanBeethoven2 6 лет назад

ILLUMINATI CONFIRMED

@numberphile 12 лет назад

@TomatoBreadOrgasm I didn't... the bowling alley actually gave us the ball and said we could keep it!

@NoriMori1992 8 лет назад

11:26 - "Pies are coming in." WHERE ARE THE PIES, GIVE THEM TO ME.

@tobiasgreven254 8 лет назад

I ALREADY ATE THEM ALL, SORRY

@89t4e784tuhdfgdfh 12 лет назад

(1) this observation was made by Gregory Galperin of Eastern Illinois University many years ago, and appeared in print in 2001. (2) The factor of 16 is independent of the fact that we use base 10. It is needed because the number of collisions between two balls required to turn the larger ball around is approximately (pi/4)*sqrt(k) where k is the ratio of masses. (3) Galperin's version does not require 16 because more collisions are counted. Google "Galperin's billiard method of computing pi".

@ajkelly451 5 лет назад

"Nothing to do with circles..." That little blip may not age well when 3blue1brown shows it has EVERYTHING to do with circles in a week or so, specifically circles in the momentum phase space. :P

@10babiscar 5 лет назад

not really to do with circles, just that you can graph a circle using a certain equation

@ajkelly451 5 лет назад

@@10babiscar If you can find pi somewhere, it probably means there is an equation for a circle involved. Also, your comment is nonsensical. If you can graph a circle using an equation (that is derived from the above thought experiment), how can you reasonably say "it has nothing to do with circles"?

@numberphile 12 лет назад

@itschampie we will

@TheRealFlenuan 10 лет назад

That didn't really explain WHY the formula was trigonometric, so the whole video was kind of pointless. He didn't teach at all why the function is the way it is; he just wrote it down and went from there… how disappointing.

@tarocchi-sciocchi 5 лет назад

I agree with you (sorry for my english). But I can explain: x = m/M x > 0 -x < x 1-x < 1+x so the fraction 1-x/(1+x) is less than 1 and it is also positive (because x < 1) in particular we can say that the fraction is a number between -1 and +1 Conclusion: the function cosine touch every number from -1 to +1, and now we can find (for each x) an angle theta with the property cos θ = 1-x/(1+x) Bye! ͜ Jo͠

@BarneySaysHi 12 лет назад

Love the fact that the video is 10 + pi minutes long, rounded to two decimals.

@GenieWeinerDog 11 лет назад

13:14 minutes long. *Claps*

@Frankie13074 4 года назад

lol 3.14 plus 10

@xXBedaXx 9 лет назад

yeah, but where does the 16 come from? Is it 2^4 or something?

@-SUM1- 9 лет назад

This video is 10 + Pi minutes long

@WitchyLadyLily 9 лет назад

+SUM1 For me it is so annoyingly close. 13:13

@zoranhacker 9 лет назад

not really, even if it were 13:14, 10+pi minutes is actually close to 13:08, yep, fun at parties

@WitchyLadyLily 9 лет назад

I think he meant 10 minutes, 3 minutes, and 14 seconds.

@-SUM1- 9 лет назад

zoranhacker I always count my minutes seconds and hours in base 60 through and through. Often I'll say there are 2:45 hours between 3 and 5:45.

@zoranhacker 9 лет назад

+SUM1 it only seems logical to do that :)

@jampozbear 12 лет назад

It's amazing what happens when one simply tries to help others giving answers.

@karlbetcher6773 8 лет назад

What happens if n=π?

@Sporkabyte 8 лет назад

I think he said that you can only use integral values of n, so π wouldn't work

@NathanTAK 8 лет назад

+Sporkabyte We're mathematicians! We don't care if it doesn't make sense to generalize!

@hikariwuff 6 лет назад

Your mom gets involved 😆

@numberphile 12 лет назад

@Sh33un we will

@liamjohnson8210 9 лет назад

He has nice handwriting.

@clemonsx90 12 лет назад

@MrFelco He could also do an experiment on something like an air hockey table using square masses. The problem with circular masses is that if they aren't lined up _perfectly_ during the next collision they are lined up even less perfectly. Hitting at 10 degrees from the central line, for example, will cause the smaller ball to miss the larger one immediately. The position uncertainty of quantum mechanics actually limits the number of collisions.

@g-gamer4747 9 лет назад

Can you actually calculate every didget of pi with this? If you have a big enough bal and playing field of course.

@g-gamer4747 9 лет назад

Jon-Kenneth Haugen Well, theoretically every is possible with the right amount of physics and mathematiks. Theretically human kind could make it's matter less dense than air and we would be floating!

@g-gamer4747 9 лет назад

Daniel Weiss If YOU can make an infinite bal they will do that.

@g-gamer4747 9 лет назад

Daniel Weiss We already know about 2.000.000 million digets

@pairot01 9 лет назад

Daniel Weiss pi has infinite digits, never repeating in the same order. If it had an end, or a repeatted string of numbers, then it could be written as a ratio of two whole numbers a/b (that's a rational number), but pi is irrational and thus it has an infinite amount of digits

@atrumluminarium 9 лет назад

In theory yes but in practice no since momentum and kinetic energy are never perfectly conserved and you cannot go to infinity since the big ball would require infinite momentum and infinite time to count them

@Pseudoradius 12 лет назад

The big ball keeps going in the same direction until it comes to a stop. That's the point where it has transferred all its energy to the small ball. It is only after that point, that the small ball gives the energy back to it. What happens here basically is the reflection of the big ball, but the small ball makes it, so the momentum doesn't shift instantaneous, but gradually, by first transferring it step by step to the small ball, then back to the big one.

@chocolatechocochoco 9 лет назад

I think this video is not good, they describe twice the physics and the results and show only some mathematical results with absolutly no link between both.. However Pi_Balls_Ed.pdf is very interesting, i think the video should tell way more about Pi_Balls_Ed.pdf

@connorskudlarek3119 9 лет назад

Math and physics are intricately linked. Real world application would result in things like friction, air resistance would become noticeable as the number of bounces become much higher, and no material has got prefect conservation. The point is a mathematical model of a physics phenomenon, with excluded variables,

@lammatt 11 лет назад

this probably makes a good story to tell in high school physics class and a good algebra exam question too. thx numberphile

@hyunwoopark9241 5 лет назад

Who came here for 3b1b?

@drokles 12 лет назад

Amazing. Can we please see the rest of the video, though? I'm very interested in seeing the derivation, and, after all, Prof. Ed prepared well for this video because he knows there will be people who are interested in this. To me skipping the derivation means skipping the good part.

@Yodovannn 10 лет назад

How did you calculate how many collisions the balls will have? You just state it without covering it with any math. This is just lame. This ain't no sci-fi show.

@W4LL37SK83R 11 лет назад

The wall is stationary. The ball does not lose any energy when it hits the wall, but when the two balls collide, their energies are both changed based on the ratio of the masses of the two. If the balls were the same size when they collide the moving one would stop and the stationary one would start moving at the same speed. But since they have different masses the large one decelerates less than the small one at each collision.

@kindpotato 7 лет назад

Very cool. I don't understand a thing about the equation he used, but I wrote a program that does all of the collisions in Python. digits = int(input("How many digits you want: ")) bigMass = 16 * (100 ** digits) bigVel = 10 smallVel = 0 i = 0 while bigVel > 0: i += 1 momentum = 2*(bigVel - smallVel)*bigMass/(bigMass+1) bigVel -= momentum / bigMass smallVel += momentum smallVel *= -1 print(str(bigVel) + " " + str(smallVel)) print(i-1)

@msungo777 11 лет назад

That's because in that ideal scenario, ball is elastic, but the wall is not, wall is ideally rigid. What actually happens in this example, why the big ball eventually stops and goes the other way, is because the little balls is essentially an "interface" between the big ball and the wall. Since it's an elastic collision, sum of kinetic energies of both balls remains the same trough the whole ordeal, so the end result is same as if the big ball itself just hit the wall on its own.

@Enlightener57 12 лет назад

This is more of a physics problem, the thing you aren't taking account of is static friction. Static friction is the reason the ball rolls instead of glides. Because the the amount of the ball that is touching the table is so small, it doesn't matter much. You need a frictionless surface to make this work completely correctly though because the balls would be slowed down by friction (not to mention the thermal energy, which is very tiny, that is added to the balls and the wall)

@MoebiusPan 12 лет назад

conservation of energy means that the initial energy of the system (the small ball) is equal to the energy after the collision; but after the collision the system is considered to be the small ball AND the big ball. Em(initial)= Em(collision) +EM(collision)

@edwarddurrans8489 10 лет назад

Does this work only using classical mechanics or would it also work if relativity equations were used?

@alpha4021 10 лет назад

I just calculated the speeds after the first collision given N=0, so x=1/16 Mu0=Mu1+mv1 by conservation of momentum Mu0^2=Mu1^2+mv1^2 by conservation of energy so make both equations be a term of u1^2 will get a new equation u0=17/32*v1 substitute it back to the equation will yield u1=15/32*v1 thus u1=15/17*u0 and v1=32/17*u0 now we check the given formula, u1=(1+1/16)*sqrt(1/16)*u0*cos(15/17) so u1=0.16876 u0 which contradicts with the result above.

@zeru2150 11 лет назад

The formula they show at 9:46 has a factor of x, which is the relation between the masses. The question was: how many collisions you need before it changes directions and they probably got something close to Pi/(x*16*10^n). Then they picked the x in such a way that it would cancel out the bottom and you would get the digits, So, in short, 16; 1600; 160000 and so on are the ones that make the answer prettier, that's all :)

@TheDrakus04 12 лет назад

It actually does have something to do with circles. The reason why cos 0=pi/2 is because he's using radians and we use radians because the angle of a full circle is 2pi, just like the circumference of the circle. Plus, a cube wouldn't roll, since its straight, only circular shapes would roll in a straight line continuously.

@11195mhr 11 лет назад

The collision with the big ball is elastic. Kinetic energy is conserved. When the small ball hits the big ball part of its kinetic energy transfers to the big ball which makes the big ball (after repeated collisions) change direction. The whole system does not lose any kinetic energy. The collision with the wall is elastic too because the wall acts as a spring, converting energy from kinetic to elastic back to kinetic with no energy dissipating. Kinetic energy is still conserved.

@11kravitzn 11 лет назад

for general x, the number of the bounce in which the big mass begins moving backward is given by the smallest integer bigger than: pi/(4*arctan(sqrt(x)))

@danielcarmi305 12 лет назад

Does this experiment in addition to having elastic collisions also have a frictionless environment? Because even if the pool tables's walls would conserve 100% of the energy while deflecting balls, I don't think it would work because of air resistance and surface friction

@drink__more__water 12 лет назад

I love this guys handwriting... looks amazing.

@jdgrahamo 11 лет назад

He says at 2:00 minutes. N in maths just means 'any number', or in this case, a number that can vary. It's like a place-holder in the equations, meaning 'insert desired number here'.

@IceMetalPunk 12 лет назад

Yes. Remember all the times he mentioned that no energy or momentum can be lost, and that's it's an elastic collision? That's exactly what that means.

@jtparm2 12 лет назад

The one thing I want to know is. What exactly is pi? What determines what the digits are and why doesn't it have an end? Much appreciation.

@henriquemalta 12 лет назад

These are great videos. Now I think you guys should do videos about e and i, the imaginary unit. I would especially like a video about Euler's formula and Euler's identity. Thank you so much.

@Phaze252 11 лет назад

Pretty much how we defined it. One primary purpose of the factorial is enumerating how many ways you can arrange objects in a set. For 2 objects, 2! = 2, for 3, 3! = 6. How many ways can you arrange zero objects (or 1 empty set) is 1. Or you can rearrange the way we express it. n! = (n-1)! * n if you substitute 1 into the expression, (1-1)! * 1 = 0! * 1 = 1. Use whichever helps you get the idea.

@JWY 12 лет назад

16 is 4 squared and if you graph the speed of either ball against the other you get a quarter circle - well, a squashed circle as the little ball speed is maximum 4 * 10 * n times the big ball's maximum speed. The speeds exchange evenly when expressed in polar coordinates corrected for the squashing. That is to say the changes are in descrete equal angle changes (angle in the quarter circles corrected for the squashing).

@LadyTink 8 лет назад

how would this be changed if you desired number of collisions was instead tau. howeverI don't know the exact process involved, and I'm just curious if anyone else knows

@Onoma314 11 лет назад

Consider the expansions of pi numerals ( 3 ) ( 3*1 ) ( 3*1*4 ) (3*1*4*1 ) ( 3*1*4*1*5 ) do 11 iterations of the multiplicative expansion, then another series with the products of the previous order of operations where the first product is the multiplier of the second, and so on. Immediately, the numbers give things like :radius of the moon, reciprocals of the speed of light, distances between planets and the sun, radius of the sun, etc, also the increments in the entire precession of the equinox

@Creaform003 12 лет назад

I wonder if this works in other base values by changing 100 to (in hex, 16X16) to give the decimals of pie in hexadecimal?

@zerocoolojoia 11 лет назад

2 questions, if I may: 1st: Does "Point of Contact" between "Ball A" and "Ball B" being RA>RB affects the number of times needed to null out the movement of Ball A? 2nd Question: Why wasn't "Friction" added/subtracted into the formula?

@WildEngineering 11 лет назад

There is a flaw in this, please correct me if I am wrong. If the ball is elastic and conserves energy, when it hits the bigger ball it will not transfer energy in to it............. so it will not slow. Also the small ball cannot receive energy from the big ball if it is elastic. IMO.

@dwarduk2 12 лет назад

Yes; it's normally taken to come under the ambiguous term "conservation of energy"; it's normally understood that friction and drag necessitate energy "loss" as heat.

@patrickleahey4574 10 лет назад

Love it and how a circles sneaks it's way in through a wave activity.

@TimelordR 12 лет назад

Finally! Some intellectual material on RU-vid!

@AussieEvonne 12 лет назад

@maxxchannel I'm glad he asks "stupid" questions, so that people like me have a shot at understanding it. He asks the questions I want to ask. Good one, Brady and Prof Ed.

@JWY 12 лет назад

I simulated the 1600x1 case and got 31 bounces to reflection. The plots of speeds describes a near perfect squashed circle, the little ball reaches nearly 40x the initial speed of the big ball (and 40x40==1600) just as the big ball reverses. The big ball doesn't stop on the reversing bounce though.

@Tupster 11 лет назад

No matter how close two points get they will never touch since they have no volume and you can always fit more points between them. There is always a scale you could zoom into two points where they look light years apart. They could become the same point, but doesn't even make any sense, since there would not be two points anymore.

@eddieboyky 12 лет назад

I have no clue what he's talking about, but I love to watch him write.

@Bob_Burton 11 лет назад

I wondered that, but it would have been nice to have it explained as it was so vital to the correspondence with the digits of pi. As it was it appeared like a rabbit out of a hat, I also wondered whether the answer depended on the balls being of the same diameter or at least colliding with their centres level with their plane of movement..

@JWY 12 лет назад

I simulated this and I'm not so confident that this worked out as cleanly as stated. It looked like the remaining speed on the large mass after 3 hits was still positive and maybe needed a 0.14ish further hit to stop - or maybe there's an "off by one" error in my understanding of this. Wow the little ball gets moving really fast after a few bounces.

@MrSaigot 12 лет назад

well since the ball does not lose energy as it moves it's exerting the same force on the large ball every time it hits, regardless of the distance either ball has traveled. The only reason the boundary is needed is too change the direction the little ball is traveling. If the boundary was further away the ball would still have the same energy and reflect back with the same energy in this situation because all energy is conserved.

@wowsa0 12 лет назад

In another base, say base 7, you will need to make changes, you'll need to make the mass 16*(49) bigger rather than 16*100 times bigger. The amazing thing is though that the answer to this problem for arbitrary masses is a formula involving pi, so chucking in a 'normal' number like a factor of 16 or 15 if you like gives you back an answer involving Pi. They deliberately chose the factor to demonstrate this in the most dramatic way possible it's true, but that doesn't make it any less amazing.

@ZantierTasa 12 лет назад

Although he went about it rather rudely, MoebiusPan was correct. It is the collisions that are said to be elastic, not the objects themselves. When the small ball collides with the big ball, their kinetic energy's have changed, but the sum of their energies remains constant. Because the big ball is slowing down, the small ball speeds up. The collision with the wall is elastic too, where no energy is transferred.

@cwjakesteel 11 лет назад

I mean, remember that there is left over energy after the balls can't touch any more. So surely on a small scale with small particles it would work. But on a large scale, the surface the ball is bouncing out of has to have huge inertia for the energy conservation to be significant.

@whytauisrightandpiiswrong3296 12 лет назад

To answer your question, I don't really know the answer, but here is my guess: Ratio:......# of times 16..........3 1600......31 When the size ratio multiples by 100, the # of bounces multiplies by ten, so I would assume this to be a square law. That if you multiply the ratio by 4, then the # of times multiplies by 2. In which case, 16*4 = 64, so if you did that with 64, 6400, 640000, and so on, perhaps you would get the digits of tau. Again, I'm not 100% sure that's right.

@assassinbbx 12 лет назад

@theschnookle and Buffons matches and sounds of Pi are 6.28/2=3.14 and the Pi with length of 9:42/3 is 3.14 Numberphile!

@NathanClingan 11 лет назад

Which is why it might work better with squares in a weightless environment, or perfectly aligned spheres. The point is not (obviously) the mechanics of the two balls, but the rate of transfer of momentum.

@N3bu14Gr4y 11 лет назад

I think the theory was referring to an abstract one-dimensional space where there was no other dimension into which energy could be transferred, without gravity or rolling friction.