2b:null 2c:["$","title",null,{"children":"How to Invent Calculus 2.1.0. | AxiomTutor | Observable"}] 31:[] 2f:{"id":"e103549de65ca5a7","slug":null,"title":"How to Invent Calculus 2.1.0.","thumbnail":"7eed361c1d9976ff36d4de4943f12d4aa1e07e32a0efe6e510280c407b862542","access_level":"public","create_time":"$D2023-09-29T13:19:52.050Z","update_time":"$D2023-09-30T03:27:04.568Z","view_time":null,"first_public_time":"$D2023-09-29T13:19:52.050Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"09c31aeb403ca87c","login":"axiom-tutor","name":"AxiomTutor","avatar_url":"https://avatars-next.observableusercontent.com/2bf1d793dac61ef22bdbbceae725f52d3b12334d88b2ecaf61864a154e830f64","tier":"starter_2024","bio":"Mathematician focused on teaching advanced mathematics in a \"reconstuctive\" way."},"version":71,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"29f8a483928726bc","login":"axiom7","name":"Axiom","avatar_url":"https://avatars-next.observableusercontent.com/ad1f41069aec7011f5912c322e8999450aaf31252cd1d9a43f1ed7551f2c4ea1","tier":null,"bio":""}],"members":[{"type":"individual","id":"29f8a483928726bc","login":"axiom7","name":"Axiom","avatar_url":"https://avatars-next.observableusercontent.com/ad1f41069aec7011f5912c322e8999450aaf31252cd1d9a43f1ed7551f2c4ea1","tier":null,"role":"editor"},{"type":"team","id":"09c31aeb403ca87c","login":"axiom-tutor","name":"AxiomTutor","avatar_url":"https://avatars-next.observableusercontent.com/2bf1d793dac61ef22bdbbceae725f52d3b12334d88b2ecaf61864a154e830f64","tier":"starter_2024","role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"titling":"auto","title":"How to Invent Calculus 2.1.0.","cells":[{"id":0,"value":"# How to Invent Calculus 2.1.0.\n## A New Theory of Gravitation\n### The Constant Acceleration Theory\n\nChapter 2, section 1, version 0\n\n---\n\n## Building a New Theory of Free Fall\n\nNow that we've rejected Aristotle's theory of free fall, we still don't know how gravity works! We need to replace the old theory with a better one. But what should our new theory be?\n\nWell for one thing, in the data that we collected to test Aristotle's theory, we can notice a pattern. \n\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":10,"value":"We could imagine that, when the object is released from a total of 3 meters, it first must fall through the distance of the first meter. It seems reasonable that the time that the object spends passing through this first meter, is the same as it does when released from 1 meter off of the ground. That is to say, when released from 3 meters, it spends 0.3 seconds falling through the first meter. \n\nContinuing in the same sort of way, consider how long it falls through the first 2 meters, which would be 0.45 seconds. Therefore the time it spends falling through the second meter only is ${tex`0.45-0.3=0.15`}. Notice that the object must be traveling at about twice the speed since it covers a meter in half the time. \n\nContinuing likewise, the time spent traveling the last meter must be the entire time, 0.55, subtracting the time of the first two meters, 0.45. Therefore it spends ${tex`0.55-0.45=0.1`} seconds traveling through the final meter. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":29,"value":"colorBox(\"AA\", calcRate)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":35,"value":"Whether any pattern stands out or not, it is clear at least that the speed is increasing. In fact the speed from the first interval (i.e. time spent traveling the first meter) is approximately ${tex`\\frac{1m}{0.3s} = 3.33...m/s`}. The speed in the second meter is approximately ${tex`\\frac{1m}{0.15s}= 6.66...m/s`}. And the speed in the third is approximately ${tex`10m/s`}.\n\nSo if we think about how the speed changes, it increases, and in fact it seems to increase *faster*. Of course we are only looking at a small sample of data points. But you could become convinced that the pattern continues as the heights are increased beyond this. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":36,"value":"Besides seeing this from the data, it also just kinda makes intuitive sense. We all know that if you fall from a greater height, you hurt more! The impact is greater. So both through the data and through some amount of physical intuition, it seems to make sense to hypothesize that the speed of objects in free fall increases. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":51,"value":"Of course that is still not a mathematical theory. Aristotle at least had an equation, and we still have a vague description. In order to be more precise it'll help to define some terms.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":47,"value":"## Position, Velocity, Acceleration\n\nWe have been measuring the height of the object in free fall, for each moment of time. This is called the **position function** -- it is a function because for each input (time) there is a corresponding unique output (height). Note that this is unlike, say, the relationships between some other quantities. \n\nFor instance the height and weight of a person are related quantities, but their relationship isn't a function. Knowledge of a person's height does not necessarily determine their weight, so that there isn't one unique corresponding value. But also there are related physical quantities which are not related by any function. For instance, consider the oscillation of a pendulum. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":69,"value":"

\n Pendulum with two points at equal height.\n

\n\n

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":12,"value":"# Behind the Curtain","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":14,"value":"import {colorBox} from \"@axiom-tutor/color-boxes\";","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":26,"value":"calcRate = md`\n### **Exercise 1**. Calculate the rates. \nFind the speed that the object travels through the first meter, then the second, then the third. (Of course because it is falling through one meter in each instance, then you'll notice that the calculuation ends up being very simple.)

\n\nIs there a pattern in these spends? Do they differ by the same quantity? Do the halve, double, or show any other noticeable pattern?

\n\n(Don't get too concerned if you can't spot a pattern just yet. We will eventually derive it rather than merely guess it.)\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":60,"value":"import {init, twodgrid, spinsphere, gui} from \"@axiom7-ws/babylonjs-demo\"","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":62,"value":"BABYLON = (await import(\"https://esm.sh/babylonjs@6\")).default","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":64,"value":"{\n const bag = init(\"pendulum\");\n const scene = bag.scene;\n twodgrid(scene);\n const sun = spinsphere(scene, 0, 0, 0.1, 1, 1, 0, 2);\n const earth = spinsphere(scene, 10, 0, 0.1, 0, 0, 1).sphere;\n const pivot = new BABYLON.TransformNode(\"root\");\n pivot.position = BABYLON.Vector3.Zero();\n earth.parent = pivot;\n const ax = new BABYLON.Vector3(0, 0, 1);\n gui(scene, bag.camera);\n\n bag.engine.runRenderLoop(() => {\n pivot.rotate(ax, 0.01, BABYLON.Space.WORLD);\n bag.scene.render();\n });\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W31","stdlib":"1"}} 30:"$2f:body"