Bungee Jump Simulation (Second order Ordinary Differential Equations) A team of engineering students is planning a bungee jumping trip. One of the preparation tasks is to write a MATLAB program to simulate high-altitude bungee jumping using a 150-meter bungee line. The purpose of the simulation is to estimate the peak acceleration, velocity, and drop distance of the jump to ensure that the arresting force of the bungee is not too great and the jump off point is high enough so that no one will hit the ground. The equation to use for the analysis is Newton's Second Law, F = ma where F is the sum of the gravitational, aerodynamic drag, and bungee forces acting on the jumper, m is the mass of the jumper (which is 70 kg), and a is the acceleration. Define the distance the jumper falls as the variable x (which is a function of time, x(t)). The jumper's velocity and acceleration are then represented as x' and x", respectively. The Newton's equation to solve for acceleration: x" = F/m Next, determines the forces making up F. The gravitational force will be the jumper's weight, which is: W = mg = (70 kg) (9.8 m/s²) = 686 N The aerodynamic drag, D, will be proportional to the square of the jumper's velocity, D = c (x')², but the value of the constant c is unknown. However, experienced jumpers know that the terminal velocity in a free-fall is about 55 m/s. At that speed, the aerodynamic drag is equal to the weight of the jumper, so c can be determined using: C = D / (x')² = (686 N) (55 m/s)² = 0.227 kg/m Finally, after the jumper has fallen beyond the bungee cord length, the slack in the bungee will be eliminated, and it will begin to exert an arresting force, B, of 10 N for every meter that it is stretched beyond 150 m. The bungee also has a viscous friction force, R, once it begins to stretch, which is given by: R = -1.5 x' Thus, there will be two regions for computing the acceleration. The first equation will be used when the distance x is less than or equal to 150 m: x"=F/m = (W-D) / m = (686 -0.227 (x')²) / 70 A second equation will be used when x is greater than 150 m: x" = F/m = (W-D-B-R)/m = (686 - 0.227 (x')² - 10 (x - 150) — 1.5 x') / 70 Simulation: • • • Create one Simulink embedded function model to simulate the bungee jumper's distance (x) vs. t, the velocity (✗') vs. t and acceleration (x") vs. t for the first 500 seconds of the jump. Adjust the Simulink model maximum step size to ensure the simulation calculates enough points to obtain the maximum and minimum values of the simulation. Use the "To File" blocks in the Simulink model to save the simulation result, distance (x) vs. t, the velocity (x') vs. t and acceleration (x") vs. t, to a .mat file. Create a MATLAB program that will automatically activate the Simulink model and run the Simulink model. Load the .mat file generated by the Simulink to the MATLAB program. Use the ode45 method to simulate the bungee jumper's distance (x) vs. t, the velocity (x') vs. t and acceleration (x") vs. t for the first 500 seconds of the jump again. Generate one figure that contains 3x2 subplots to plot and compare the Simulink solution and the ode45 solution side-by-side. The figure should include the following: о ○ Plot the bungee jumper's distance (x) vs. t chart with the Simulink Solution on the left and plot the x vs. t chart using the ode45 solution on the right. Plot the bungee jumper's velocity (x') vs. t chart with the Simulink Solution on the left and plot the x' vs. t chart using the ode45 solution on the right. о Plot the bungee jumper's acceleration (x") vs. t chart with the Simulink Solution on the left and plot the x" vs. t chart using the ode45 solution on the right. Title and label plots clearly. Answer the following simulation analyses questions and print answers on screen or to a text file. • о What is the estimated peak acceleration value of the entire jump? 000 What is the estimated peak velocity of the jump? What is the estimated maximum drop distance of the jump? (How far will the jumper fall before he starts backup?) The bungee jump simulation starts at 0 seconds. How many seconds will the jumper fall to reach the maximum drop distance? From the simulation results, determine the velocity and the time when x = 150. This is the point at which the slack in the bungee is eliminated. о о о How high should the lowest jump off point be to ensure a safety factor of 2?