## Assignment 17: ODEs - Projectile Motion II

Instructions: In Assignment 12, we considered projectile motion using Excel spreadsheets as the computational and graphing tool. In this assignment, you will repeat the solution of the projectile motion problems, but you are to use MATLAB as the computational tool. There is a MATLAB code in Garcia called balle.m (listing given on page 58) that can be modified to handle the problems given below. However, if you wish to write your own code, you may do so, or use balle.m as a model for writing your own code. A discussion of the physics of this problem is given in Physics Application: Projectile Motion.

#### Problem 1. Simple Projectile Motion

Using MATLAB, set up the numerical simulation in Eq. 3 of the Projectile Motion document, and entitle your spreadsheet "Projectile Motion: Trajectory of a Cannon Shell." Perform the calculations using an initial position of zero and an initial velocity of 100 m/s with launch angles of 30o, 35o, 40o, 45o, 50o, and 55o. Graph your results with all launch angles on the same graph and post the graph on your web site. Carry out the computation for each launch angle to the point that the shell has clearly struck the ground. Through interpolation, find the velocity and the horizontal position at the moment of impact. Compared your numerical results from the MATLAB calculation with those from Excel and comment.

#### Problem 2. Bicycle with Air Resistance

Using MATLAB, set up the numerical simulation for the motion of a bicycle, which is called "Bicycle Racing: The Effect of Air Resistance." Perform the calculations without air resistance using Eq. 3, where there is no vertical motion only horizontal, and with air resistance Eq. 8. For the parameters of the problem, use initially the following. Assume the initial velocity of the bicycle is 4 m/s (about 10 mph), the power input by the rider is 400 W, the drag coefficient is 0.5, the cross-sectional area of the bicycle and rider is 0.33 m2, the mass of the bicycle and rider is 70 kg, and the density of air is 1.29 kg / m3. For the initial time step use 0.10 seconds. Calculate the velocity as a function of time for no drag and with air resistance. Graph your results on the same graph and post on your web site. What is the terminal velocity with air resistance for the given parameters? Compared your numerical results from the MATLAB calculation with those from Excel and comment.

#### Problem 3. Cannon Shell with Air Resistance and Density Variations

Using Excel, set up the numerical simulation of the motion of a cannon shell, which is called "Projectile Motion: Trajectory of a Cannon Shell with Air Resistance." Perform the calculations with air resistance (Eq. 11), and with air resistance and a density correction (Eq. 14). As parameters, use B2 / m = 4 x 10-5 m-1, an initial velocity of 700 m/s, launch angles of 35o and 45o, and the atmospheric scale height h = 1.0 x 104 m. Graph your results on one graph and post on your web site. Compared your numerical results from the MATLAB calculation with those from Excel and comment.

Physics & Astronomy Department, George Mason University
Maintained by J. C. Evans; jevans@gmu.edu