Day-18_Pre-Class_ModelingWithODEs-STUDENT - Jupyter Notebook

3/17/24, 11 : 37 PM Day-18_Pre-Class_ModelingWithODEs-STUDENT - Jupyter Notebook Page 1 of 16 http://localhost:8888/notebooks/Downloads/Day-18_Pre-Class_ModelingWithODEs-STUDENT.ipynb Day 18: Pre-class Assignment: Modeling with Ordinary Di ff erential Equations Put your name here Goals for today's pre-class assignment In this assignment, you will: Create a model for a skydiver falling using di ff erential equations Setup a system of di ff erential equations (i.e., more than 1). Solve these systems of di ff erential equations using the update method from the previous assignment. Solve these systems of di ff erential equations using solve_ivp We will build o ff the previous assignment where we introduced using update equations to numerically integrate a (mathematical) function. Here we will expand our tool kit by numerically solving multiple di ff erential equations simultaneously, and we'll solve them using a tool built into SciPy called solve_ivp . But first, let's talk about the role of di ff erential equations in modeling.

3/17/24, 11 : 37 PM Day-18_Pre-Class_ModelingWithODEs-STUDENT - Jupyter Notebook Page 2 of 16 http://localhost:8888/notebooks/Downloads/Day-18_Pre-Class_ModelingWithODEs-STUDENT.ipynb The Role of Di ff erential Equations in Computational Modeling We often use equations in computational modeling, and many of the systems we wish to model contain things that are changing : changing in time (E.g., Temperature or heart rate) changing in space (E.g., Temperature or elevation) changing strategies (E.g., deciding whether to invest) changing numbers (of people, molecules, stars, etc.) changing prices and so on... In mathematics, we describe change with calculus and derivatives in particular. When an equation has a derivative in it, it is called a " di ff erential equation (https://en.wikipedia.org/wiki/Di ff erential_equation) " (as opposed to, say, an algebraic equation). The types of equations can be used to model many di ff erent systems, and in this assignment, we'll see how we can use them to model the free-fall of a skydiver. Question : What is a value or system that you deal with on a daily basis that is changing? A value that I deal with is money.

3/17/24, 11 : 37 PM Day-18_Pre-Class_ModelingWithODEs-STUDENT - Jupyter Notebook Page 3 of 16 http://localhost:8888/notebooks/Downloads/Day-18_Pre-Class_ModelingWithODEs-STUDENT.ipynb Modeling a falling skydiver without air resistance In this assignment, we will be modeling the motion of a skydiver. There are two variables that will be changing that we need to consider: The height of the skydiver (I.e., how far above the ground they are). This value will change with time as the skydiver gets closer and closer to the ground during their fall. The speed (or, really, the velocity ) of the skydiver * (I.e., how fast they are falling towards the ground). The longer the skydiver falls, the faster they will be falling. Since we have two values that are changing, we will need two di ff erential equations. We can express these two values using update equations: The main variables to know are: refers to height, and refers to velocity. Note 1: , meaning that velocity is defined as the derivative of height. Note 2: For the first part of this problem, , where is a constant equal to . (Everything on Earth picks up speed at roughly the same rate, as long as we ignore air resistance. (https://www.youtube.com/watch?v=frZ9dN_ATew) ) = + new old = + new old = = - 9.81 The derivs Function In order to solve these problems numerically, we need to specify what and are. We’re going to use a derivs function, just like we did in the previous assignment, except this time it will return the derivatives for two equations. Specifically, we want it to return: Putting it into code, we get the following: = =

3/17/24, 11 : 37 PM Day-18_Pre-Class_ModelingWithODEs-STUDENT - Jupyter Notebook Page 4 of 16 http://localhost:8888/notebooks/Downloads/Day-18_Pre-Class_ModelingWithODEs-STUDENT.ipynb In [1]: The derivs function is the heart of how we will solve di ff erential equations. (It may seem silly to define a function that takes in two values and returns the same two values, but we'll be making it more complex in a second.) Initial Values We first will begin by assuming: meters meters/second gravity: meters/second^2 falling duration: seconds timestep = 3000 0 = 0 0 = - 9.81 = 30 = 1 Running the Model This code numerically solves for the motion of the skydiver and plots the result. It should be similar to the code you wrote in your previous assignment. Review each line of code an make sure you understand what it's doing and then execute the code. In [2]: # Create a function to compute derivatives of velocity and height def derivs (v,g): # derivative of height is velocity dhdt = v # derivative of velocity is gravity for this example dvdt = g return dhdt, dvdt # Import modules import matplotlib.pyplot as plt # Initialize variables h = 3000 # initial height: m v = 0 # initial velocoty: m/s g = - 9.81 # gravity: m/s^2 t = 0 # initial time tmax = 30 # Falling time dt = 1 # timestep

3/17/24, 11 : 37 PM Day-18_Pre-Class_ModelingWithODEs-STUDENT - Jupyter Notebook Page 5 of 16 http://localhost:8888/notebooks/Downloads/Day-18_Pre-Class_ModelingWithODEs-STUDENT.ipynb dt = 1 # timestep # Initialize lists for storing data height = [] velocity = [] time = [] # Append initial values to lists height.append(h) velocity.append(v) time.append(t) # Create a time loop that will update the skydiver over time # Use a while loop that will loop until t > tmax while t <= tmax: # Compute derivatives for use in update equations dhdt, dvdt = derivs(v,g) # Update Equations h_new = h + dhdt * dt # new height v_new = v + dvdt * dt # new velocity # Append values to height and velocity lists height.append(h_new) velocity.append(v_new) # Update old height/velocity with new height h = h_new v = v_new # Increase time t += dt # t = t + dt # Update time list time.append(t) # Plotting height/velocity vs time plt.figure( 1 ) plt.plot(time, height, color = 'green' ) plt.xlabel( 'Time [seconds]' ) plt.ylabel( 'Height [meters]' ) plt.grid() plt.figure( 2 ) plt.plot(time, velocity, color = 'purple' ) plt.xlabel( 'Time [seconds]' ) plt.ylabel( 'Veloctiy [meters/second]' ) plt.grid()