Grandma goes for bungee jumping From Newton\'s second law of motion, the rate of

Question

Grandma goes for bungee jumping From Newton's second law of motion, the rate of change of the jumper's velocity with respect to time can be written as dt where v-downward vertical velocity (m/s), t = time (s), g = the acceleration due to gravity, cd- drag coefficient (kg/m), and m = the jumper's mass This is a differential equation, and calculus might be used to obtain an analytical solution for v as a function of t. However, numerical solutions can be obtained by using iterative approaches. For example, use Euler's method, an iterative scheme for the velocity can be written as dv dt ut A pseudo-code for a uniform step size 4twill look like this Define f(r, v) = g-_vlvt 1. 2. 4. im Input to and vo. Input step size, 4, and the number of steps, n for j from 1 to n do a. z=f(to, vo) d, to=t1 5. end Submit a report for grade. Your report should include the following items among other things Problem statement. A MATLAB program that used to perform the calculations. Your program should use a loop as the basis for the algorithm, and contain sufficient comments. Calculation of a jumper's velocities for the first 20 seconds. Plot velocity vs time. Use the following parameters: g = 9.81 m/s2, c.-0.25 kg/m, and m = 60 kg. Investigation of the effect of drag coefficient on velocity by using cd-0.25, 0.50, 0.75 kg/m, and plot the results on one graph. Investigation of the effect of the jumper's mass on velocity by using m-40, 60, 80 kg, and plot the results on one graph. Discussion of the issues such as jumper's maximum speed, strength required for a bungee cord, etc

Explanation / Answer

%Chegg#4Part1

%Given
g = 9.81;
Cd = 0.25;
m = 80;

syms v ; %Let v be a symbol

f = @(v) g-(Cd/m)*v*abs(v); % A Function handler defined for the given dv/dt

%Initial Condition
current_time = 0 ; current_velocity = 20 ; %Assuming a initial velocity as 20 m/s, you can edit later too

%Initial time step
del_t = 0.1;
%Number of steps
n = 200;

%Preallocating variables
temp_velocity = zeros(n+1,1);
%Inserting the initial velocity
temp_velocity(1) = current_velocity;

for j = 1:n

z = f(current_velocity);
  
temp_velocity(j+1) = current_velocity + (z*del_t); %Euler's Rule
  
temp_time = current_time + del_t; %Incrementing time
  
current_velocity = temp_velocity(j+1); %Changing current_velocity value for the next loop
  
current_time = temp_time; %Changing current_time value for the next loop
  
end

%Defining time axis for the plot
x_time = [0:0.1:20];

plot(x_time,temp_velocity);

The Above is a single .m file, which will plot for a single value of Cd and mass . The Below is for Varying Cd, Varying Mass. Look out to comment and uncomment properly. The code will work for either varying mass or varying Cd at a single run.

%Chegg#4Part2

%Given
g = 9.81;

%If varying mass, constant Cd
%Uncomment below two lines
%Cd = 0.25; %Constant Cd
%m = [40,60,80] ; %Varying mass

%If varying Cd, constant mass
%Uncomment below two lines
%m = 60; %Constant mass
%Cd = [0.25;0.5;0.75]; % Varing Cd

%The code will work only for either one condition at a time

syms v ; %Let v be a symbol

%Initial Condition
current_time = 0 ; current_velocity = 20 ; %Assuming a initial velocity as 20 m/s, you can edit later too

%Initial time step
del_t = 0.1;
%Number of steps
n = 200;

%Preallocating variables
temp_velocity = zeros(n+1,3);
%Inserting the initial velocity
temp_velocity(1) = current_velocity;
temp_velocity(2+n) = current_velocity;
temp_velocity(3+n+n) = current_velocity;

initial_velocity = current_velocity;
initial_time = current_time;
%Varying Mass
for i = 1:3
%If Variation with Cd,uncomment below two lines
%Actual_Cd = Cd(i);
%f = @(v) g-(Actual_Cd/m)*v*abs(v);
%If Variation with mass,uncomment below two lines
%Actual_m = m(i);
%f = @(v) g-(Cd/Actual_m)*v*abs(v);
  
for j = 1:n

if j==1
z = f(initial_velocity);
temp_velocity(j+1,i) = initial_velocity + (z*del_t); %Euler's Rule
temp_time = initial_time + del_t; %Incrementing time
else
z = f(current_velocity);
temp_velocity(j+1,i) = current_velocity + (z*del_t); %Euler's Rule
temp_time = current_time + del_t; %Incrementing time
end

current_velocity = temp_velocity(j+1,i); %Changing current_velocity value for the next loop

current_time = temp_time; %Changing current_time value for the next loop
  
end
  
end

%Defining time axis for the plot
x_time = [0:0.1:20];
hold on;
plot(x_time,temp_velocity(:,1));
plot(x_time,temp_velocity(:,2));
plot(x_time,temp_velocity(:,3));
hold off;

Key Take away :

1. More the mass of the jumper, more higher velocity he can attain (See graph plots)

2. More the coefficient of drag, lesser is the velocity that the jumper can attain.

3. Jumper's maximum velocity stables after a particular amount of time.

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