Math 153 Module 2 Applied Problem Background: Calculus was invented by Sir Isaac Newton largely to explain Keplers Laws of Planetary motion. These laws were based on observations of planets and, as such, Keplers Laws were subject, of course, to a rather serious problem: What if there were other planets out there that didnt follow his observations? Kepler was using ancient technology (1600s) and could only see a few planets. Newton formulated two laws of physics (force is mass times acceleration and the strength of the gravitational force between objects is inversely proportional to the distance between them and proportional to the product of their masses) and used his new calculus to derive Keplers Laws mathematically. They perfectly describe the motion of all the planets, asteroids, comets, and other bodies. Except for the ones they dont. And those discrepancies lead to the discovery of special relativity (Einstein) and dark matter (multiple people). The Problem: On January 21, 2020 an asteroid is detected approaching Earth and you are tasked with determining whether or not the asteroid will impact the planet. If it will impact the Earth, you are to estimate the kinetic energy of the impact in order to determine potential damage. To facilitate the calculations, you will make the following assumptions. You will approximate the orbit of the Earth with a perfect circle. In actuality, the Earth has a very slightly elliptical orbit but we will ignore that in this project. You will assume the period of Earths orbit is exactly 365 days (it is actually about 365.25 days). You will assume the Sun is the center of the Earths orbit (it is close to the center) and that the radius of ). To make the equations more manageable, you will use a three dimensional (x,y,z) coordinate system with the Sun at the origin and the Earth orbiting in the (x,y,0) plane. In this coordinate system, the Earth will follow a parametric equation around a circle of radius 1 AU, of period 365, starting at (1,0,0) and travelling counterclockwise (standard parameterization). In this coordinate system, the asteroid is located at (2.2, 4.5, 1.6) in AUs on January 21, 2020. It is located at (2.042088,4.194329,1.462857) AU 6 days later. You will assume that will travel along a straight line trajectory. . It actually is more of an ellipsoid, but only slightly. Note that youll need to convert this to AU at some point. . By velocity, we of course mean speed, which is the magnitude of the velocity vector. Note that if there is an impact, you should convert the speed to the standard meters per second. Although you can approach the problem in a variety of ways, here is one path forward (not the only one). First understand the problem. Figure out where thing are located and how you might approach the problem if this wasnt here. Seriously. Spend time thinking about this before you start. Find the parametric equation of the Earths orbit. This should be easy since it is just a circle in the xy plane of 3 space. Then, find the parametric (or vector) equation of motion for the asteroid. You are given its position at time t=0 and at time t=6. This should be straightforward. The t=6 is important here so make sure you take this in to account. You should assume the asteroid is travelling at a constant speed. You need to determine if the asteroid strikes the Earth. While you could try to solve to see if your orbit equation and your asteroid equation are ever in the same place at the same time, this would actually be solving for if the asteroid hits the center of the Earth. But the Earth isnt a point, it is a sphere with a given radius. So, you could, instead, find the distance from the Earth center to the asteroid as a function of time, t (use the distance formula you learned). Then, you could solve for if the distance is ever less than or equal to the radius of the Earth (in AU, not km). If the asteroid impacts the Earth, you will need to figure out the impact speed. To do this, find the magnitude of the velocity vector from the asteroids trajectory. Note that if there is an impact, you should convert the speed to the standard meters per second. Key Points: Note that although many of the steps are straightforward, there is a challenge toward the end. Dont even think about trying to do the solve by hand. You will most likely find that your TI calculator is overwhelmed. When I tried WolframAlpha, it wasnt happy either. You may have more luck plotting the distance function and then estimating when (if ever) the distance drops below the radius of the Earth (again, in AU). Try not to round, even a little. Carry full decimals. Because the scales dont match up well (the Earths radius in AU is very small), if you round you may get enough accumulated error to thrown off the answer. Note that if there is an impact, you should convert the speed to the standard meters per second. It seems like there are many assumptions being made. In fact, most of these are pretty mild in nature and are made to make the problem a little more manageable. Partners: You can work with a student in our course (or the other section I am teaching) on this problem if you choose to. You are not required to work with another student. You can work with different students on different problems during the quarter. You are encouraged to complete at least one applied problem on your own. Required Material: Please read the link in the Background section and complete Module 2, Assignment 11. A Point to Think About but to do Nothing With: Once you leave school and start to work professionally, it is unlikely that you will be allowed to do math by hand, except for rudimentary calculations. Even in school, your professors will expect that you have some facility with some computational packages. Structured programming languages like C/C++/C# or Java are general purpose languages and are probably not what you really want to learn (despite most programs requiring an introductory Java or C class). It takes too long to code things like this project in Java. Symbolic computational programs like Mathematica (the Wolfram Language) allow for symbolic manipulations and are extremely powerful. More numerical type programs such as MATLAB are very, very common in engineering fields and you will be expected to know it, even if you havent had a class in it (we teach it in ENGR240). I developed and then solved this problem using a programming language called R but could just as easily have used Mathematica, Maple, or MATLAB. I used R because there is solid support for it, it has a free distribution, and Ive been using it for various machine learning and data science projects Ive been working on. It is extremely important that you learn some sort of computational programming package such as Mathematica or MATLAB, and ideally a language or two such as R, Python, Java, or C. You should start to use these in your school work in order to build some facility with them. It will give you an edge in course work, applications to internships and lab positions, and start you down the road to a modern approach to how STEM functions in the real world. The Report: Your report should include the following at a minimum: A well done introduction and conclusion that would satisfy a member of the English faculty! Explain how you found the Earths orbit and the trajectory of the asteroid, including a well integrated explanation of the assumptions made (dont just re-type mine). If you take the route outlined above, please show how you solved the distance equation (i.e, determined impact or not). You dont need to given a long explanation. For example, you could generate a plot of the distance function via Desmos or other and estimate the time of impact, if it exists. Or if you happen to have access to Mathematica and used the FindRoot function, please write about that. You must include one informative graphic. This is up to you, but it has to be helpful. If you used a graphical solve, you could include a well labeled graph. Or you could create a graphic showing the orbit and the start of the trajectory. You can use graphics you find on the web, but make sure you attribute them. If there is an impact, make sure you compute the kinetic energy using standard units (meters per second and kilograms). The asteroid is moving very fast. There is no minimum length, but I doubt you can get this done in less than four pages. All mathematical symbols must be typeset and look good. Do not write something like sqrt((9-x)^2- Your end goal is to have something that you would feel comfortable handing in to SpaceX when you apply for an internship and they ask for a writing sample. Or for a lab job working with gravity wave detecting lasers. Dont just think of this as a grade in a class: Students have used projects from my classes to help get prestigious internships and lab positions. Due Date: The latest turn in date is Monday, March 15th at 11:59 pm. However, please do not wait that late to turn it in. In order to take the module 2 exam you have to turn this in first. Formatting: You must submit your report in MS Word format (.docx) to me via email (cwillett@tacomacc.edu). Be aware that if you write the file in something like Google Docs and then convert to MS Word many of the math symbols will not be translated correctly and your report will not look good. You can get free access to MS Word as a student at TCC. When I receive the report, I will send you an acknowledgment via email. The file should be named using the following format: first initial last name_app2.docx. As an example, my report would be cwillett_app2.docx. If you work with a partner, submit just one file and include both names in the file name separated by an _. If I worked with Jon Armel, I would name the file cwillett_jarmel_app2.docx. Grading: I will grade your report on a 10 point scale. The correct mathematics will get you 6 of the 10 points. The quality of your writing and your graphic(s) will get you 3 of the points. The correct formatting will get you the remaining 1 point. The Honor System: When you submit the report, you will be certifying that it is your work and that you have not plagiarized the work of another. Please cite sources of information that you are using. When you receive the graded report back, please do not distribute the graded work to others or to discuss anything other than generalities. I want you to get good feedback from me to improve future efforts.