Using Geometric Algebra to Solve Projectile-Motion Problems: An Introduction

Transcript

welcome to this latest video in our series on doing things of geometric algebra and in it we will introduce a document that is linked in the video description on how hus treat use a geometric algebra to treat constant acceleration motion now a lot of you may be uh familiar with hen's book New Foundations for classical mechanics and it is from this book that uh we're taking his treatment or expanding upon it for uh constant acceleration motion so as explained in the abstract of the document this will be the first installment in a guide to his treatment and specifically we will present a more detailed version of hess's solution to the problem of finding the time and distance at which an object or a projectile will cross a given line of sight now what do we mean by that well we can see that in this document or in this geometric algebra construction rather an object is launched with a certain initial Direction a certain initial velocity and within a certain in this case gravitational field and or a gravitational field of a certain strength and we want to know at what point the object will cross this given line of sight here so this is what we want to find out now back to our document one of the things that hus points out in his book is that to for students to get through it they need a certain amount of judicious guidance and that's what we hope to provide in this document and the document is intended to be understand Able by students and teachers who are familiar with the basics of geometric algebra and the document begins with a review of what we will see in in within the document quite a detailed review of the ideas that we will use and then the differential equations of motion for constant acceleration and also their Solutions and the solution uh H hus treats those in the form or in a visual form that's known as a photograph which presents the uh situation and its solution in terms of uh vectors of um velocity and rather than explicitly in terms of vectors of position and this this idea has some advantages so then we get into the solution of our particular problem and because this introduction is intended for high school level students or first year college there's a word of encouragement here that don't let the odd-looking equations and expressions scare you because we'll show how they can be translated into other forms that are more convenient the oddl looking equations and such are really quite uh useful for uh finding the solution but once we get to a solution in terms of these oddl look Expressions we can turn those expressions into things that are more familiar quite easily so the solution Begins by finding the time of flight uh to or at at which uh the object crosses a line of sight and the derivation is quite detailed or it's not difficult but a lot of details are provided again for people who are kind of just getting into geometric algebra and there are some little clarifications and reminders provided and also a couple of Sanity checks to find out okay are the units correct are we finding units of time or and are we do we have the correct algebraic sign so when we're all done here with that part of it we end up with a nice equation for the amount of time or or the time at which the object crosses a line of sight then this solution for time is used in finding the distance are this distance and again quite a few details are provided for the convenience of the reader and a couple of little explanations as to what's going on in each step and we end up at the end here sorry we end up here with our solution and we compare that to H's solution and the thing that's noteworthy here is that we can express this quite conveniently in terms of the magnitudes of these two bi vectors but hen has gives his solution in terms of the inner product the dotproduct of those bi vectors now that's curious because that's kind of an artificial um change from this version to that so the question is why does hon has go to the trouble of converting to this kind of a or converting kind of artificially to this version well the answer is that this version in the form of an inner product is a version that he needs for his next uh part of or his next um the next the subject that he treats next which will also be the subject the next uh video in this series and of course the treatment ends here with a with a geogebra worksheet that will allow the user to or it will be linked in the video description or the document rather that will allow the reader to check the solution for himself or herself as we have here so the reader will be able to check the solution numerically and verify that all this works and you'll be able to vary the initial velocity the gravitational strength and also the initial direction of launch and the line of sight that you're trying to find the solution for so I hope this document will be useful and I also hope that you'll consider joining our LinkedIn group pre-university geometric algebra thank you for your time and I look forward to um reading your comments