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 xx$($t$)$ x$^(\text{'})$ and x$^(\text{'}\text{'}),$ respectively. The Newton's equation to solve for acceleration:

x$^(\text{'}\text{'})=($F$)/($m$)$

Next, determines the forces making up F$.$ The gravitational force will be the jumper's weight, which is:

W$=$mg

$=(70$kg$)(9\mathrm{.}8($m$)/($s$^(2)))=686$N

The aerodynamic drag, D $,$ will be proportional to the square of the jumper's velocity, D$=$c$($x$^(3))^(2),$ 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$^(\text{'}))^(2))$

$=(686$N$)/((55($m$)/($s$))^(2))$

$=0\mathrm{.}0227$k$($g$)/($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$=-1\mathrm{.}5$x$^(\text{'})$

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$^(\text{'}\text{'})=($F$)/($m$)=($W$-$D$)/($m$)=(686-0\mathrm{.}227($x$^($N$))^(2))/(70)$

A second equation will be used when x is greater than $150$ m :

x$^(\text{'}\text{'})=($F$)/($m$)=($W$-$D$-$B$-$R$)/($m$)=(686-0\mathrm{.}227($x$^(\text{'}))^(2)-10($x$-150)-1\mathrm{.}5$x$^(\text{'}))/(70)$

Simulation:

Create one Simulink embedded function model to simulate the bungee jumper's distance $($ x $)$

vs$.$ tx$^(\text{'})$ tx$^(\text{'}\text{'})$ tx$^(\text{'})$ tx$^(\text{'}\text{'})$ 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 ode$45$ method to simulate the bungee jumper's distance $($x$)$ vs$.$ tx$^(\text{'})$ tx$^(\text{'}\text{'})$ t for the first $500$ seconds of the jump again.

Generate one figure that contains $3\backslash$times $2$x x vs$.$ tx$^(3)$ x$^(\text{'})$x$^(\text{'}\text{'})$ x$^(\text{'}\text{'})$ vs$.$ t chart using the ode$45$ 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.

What is the estimated peak acceleration value of the entire jump?

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 Add lots of comments. Make this simulation a demo worthy program for interviews. Add lots of comments. Make this simulation a demo worthy program for interviews.$|$

$($The figure should look like the following example.$)$

I need hlep on writing code on matlab and create simulink.