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Video instructions and help with filling out and completing Form 2220 Compute

Instructions and Help about Form 2220 Compute

You know that feeling you get when you have two mirrors facing each other and it gives the illusion of there being an infinite tunnel of rooms or if they're at an angle with each other it makes you feel like you're a part of a strange kaleidoscopic world with many copies of yourself all separated by angled pieces of glass what many people may not realize is that the idea underlying these illusions can be surprisingly helpful for solving serious problems in math we've already seen two videos describing the block collision puzzle with its wonderfully surprising answer big block comes in from the break lots of cuts the total number of clacks looks like pi and we want to know why here we see one more perspective explaining what's going on where if the connection to PI wasn't surprising enough we add one more unexpected connection to optics but we're doing more than just answering the same question twice this alternate solution gives a much richer understanding of the whole setup and it makes it easier to answer other questions and fun side note it happens to be core to how I coded the accurate simulations of these blocks without requiring absurdly small time steps in huge computation time the solution from the last video involved a coordinate plane where each point encodes a pair of velocities here we'll do something similar but the points of our plane are going to encode the pair of positions of both blocks again the idea is that by representing the state of a changing system with individual points in some space problems and dynamics turn into problems in geometry which hopefully are more solvable specifically let the x coordinate of a 2d plane represent the distance from the wall to the left edge of the first block what I'll call d1 and let the y coordinate represent the distance from the wall to the right edge of the second block what we'll call d2 that way the line y equals x shows us where the two blocks clack into each other since this happens whenever d1 is equal to d2 here's what it looks like for our scenario to play out as the two distances of our blocks change the two-dimensional points of our configuration space move around with positions that always fully encode the information of those two distances you may notice that at the bottom there it's bounded by a line where d2 is the same as the small blocks width which if you think about it is what it means for the small block to hit the wall you may be able to guess where we're going with this the way this point bounces between the two bounding lines is a bit like a beam of light bouncing between two mirrors the analogy doesn't quite work though in the lingo of optics the angle of incidence doesn't equal the angle of reflection just think of the first collision a beam of light coming in from the right would bounce off of a 45-degree angled mirror this x equals y line in such a way that it ends up going straight down which would mean that only the second block is moving this does happen in the simplest case where the second block has the same mass as the first and picks up all of its momentum like a croquet ball but in the general case for other mass ratios that first block keeps much of its momentum so the trajectory of our point in this configuration space won't be pointed straight down it'll be down into the left a bit and even if it's not immediately clear why this analogy with light would actually be helpful and trust me it will be helpful in many ways run with me here and see if we can fix this for the general case seeking analogies in math is very often a good idea as with the last video it's helpful to rescale the coordinates in fact motivated by precisely what we did then you might think to rescale the coordinates so that X is not equal to d1 but is equal to the square root of the first mass m1 times d1 this has the effect of stretching our space horizontally so changes in our big blocks position now results in larger changes to the x coordinate itself and likewise let's write the y coordinate as square root of m2 times d2 even though in this particular case the second mass is 1 so it doesn't make a difference but let's keep things symmetric maybe this strikes he was making things uglier and kind of a random thing to do but as with last time when we include square roots of masses like this everything plays more nicely with the laws of conserving energy and momentum specifically the conservation of energy will translate into the fact that our little point in the space is always moving at the same speed which in our analogy you might think of meaning there's a constant speed of light and the conservation of momentum will translate to the fact that as our point bounces off of the mirrors of our set up so to speak the angle of incidence equals the angle of reflection doesn't that seem bizarre in kind of a delightful way that the laws of kinematics should translate to laws of optics like this to see why it's true let's roll up our sleeves and work out the actual math focus on the velocity vector of our point in the diagram it shows which direction it's moving and how quickly now keep in mind this is not a physical velocity like the velocities of the moving blocks instead it's a more abstract rate of change in the context of this configuration space whose two dimensions worth of possible directions encode both velocities of the block Music the X component of.

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