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Application 11: Parachute Simulation 4ENGR 1221 Designed to test skills with: simulation. This assignment may be done in two-person teams. You may divide the labor as you see fit, but our recommendation is that one person complete the main simulation code, while the other writes the supporting functions (gravitational acceleration, atmospheric model, drag). Each person should submit, but please indicate your partner’s name to save the graders from duplicating effort. Overview In 2012 Felix Baumgartner set the world record for skydiving by falling from 36,402.6 meters. He reached a high speed of 377.12 m/s during his fall. This jump represents the first time an individual in freefall broke the sound barrier (308.46 m/s). Your task is to simulate his fall in Matlab using the gathered knowledge from the previous tasks. The simulation will be fundamentally similar to previous assignments, with the following additional features: · The altitude of this jump is too great for the acceleration of gravity to be assumed to be constant. You will write a function to calculate g as a function of altitude. · You will use an empirical atmospheric model to calculate the pressure as a function of altitude, rather than interpolating as was done previously. · After you get the baseline simulation working, you will adjust the drag coefficient iteratively to match the experimental result. Deliverables 1. Main script file based on the previous tasks with the changes listed in the next section. 2. Function files for the following calculations: a. Cross-sectional area vs. Time (from Homework 9) b. Gravitational acceleration vs. altitude c. Acceleration due to drag (this function should accept the drag coefficient, density, area, velocity, and mass) d. Atmospheric values (may be 3 separate functions or a single function with 3 outputs) for i. Temperature ii. Pressure iii. Air Density 3. Graphs of: a. Height, velocity, and acceleration during the jump (Note: this is a 1-D simulation: vertical motion only) b. Gravitational coefficient, temperature, pressure, and air density during the jump (use altitude as the y-axis for all 4) Data and Code Changes Use the following parameters: 1. Mass of person and suit: 118 kg 2. Starting height: 36,406.5 m 3. Parachute deployment altitude: 1524 m (Note: the parachute no longer deploys after 60 s) 4. Freefall area radius (initial radius): 0.5 m 5. Canopy area radius (final radius): 3.8 m 6. Time for parachute to open: 8 s 7. Air density: use the provided formula to calculate at each height (in the next sections) (Note: you no longer interpolate) 8. Gravitational coefficient: use the provided formula to calculate at each height (Note: remember which direction gravity points) 9. Present the time of chute deployment (must be determined automatically by your code) using an fprintf statement or message box 10. Do your initial coding with a drag coefficient of 0.5. Once you are confident that the simulation is working properly, incorporate the algorithm to optimize the drag coefficient to match the real-world result. Your code should iteratively find a better coefficient so that the parachutist’s absolute maximum speed is within 0.01 of 377.12 m/s (Figure 1). There are various ways to structure your code to achieve this, including (but not limited to) these possibilities: a) Make your main simulation a function that takes the drag coefficient as an input argument and returns the maximum speed as an output argument. Call this function repeatedly from a script that contains your optimization algorithm. b) Incorporate an additional while loop into your main script – surround the simulation code with this second loop, with the drag coefficient being changed for each iteration. c) Make a GUI with a slider, and adjust the slider until the target result is achieved. Air Density, Pressure, and Temperature · Definitions ρ = density (kg/m3) p = pressure (kPa) T = temperature (˚C) h = altitude (m) · Density based on the ideal gas law (for all h) · For h > 25,000 m · For 25,000 m > h > 11,000 m · For 11,000 m > h Gravitational Coefficient · Definitions g = gravitational coefficient (m/s2) G = gravitational constant = 6.674x10 -11 (m3/(kg*s2)) Mearth = 5.97x1024 (kg) rearth = 6.38x106 (m) h = altitude (m) · Calculation (for consistency with previous assignments, g is given as a negative number) Approximation of Drag Coefficient (Cd) Once the baseline simulation is working correctly, you will adjust the drag coefficient to match the experimental result – specifically, the maximum speed of 377.12 m/s. Figure 1: Logical diagram of change in Cd The specific algorithm that you use for adjusting Cd is up to you. A few possibilities include · Bisection (a.k.a. binary search). · Set upper and lower limits on Cd. · Set the approximation of Cd equal to the mid-point between the limits · If Cd is too high (i.e. the max velocity is too low), reduce the upper limit; otherwise, increase the lower limit. · Repeat until the difference between the upper and lower limits is negligible · Use the built-in function fminsearch · Grahical approach using a GUI with a slider