Bungee Velocity Analysis

Develop a MATLAB program utilizing Euler's formula from Chapter $1$ to numerically solve the differential

equation describing the velocity of a free$-$falling bungee jumper. Your program should plot the

approximate solution obtained from Euler's formula and compare it with the exact $($analytical$)$ solution

for the same scenario.

Differential equation: $\frac{dv}{dt}=g-\frac{c_d}{m}{v}^2$

Details:

Dependent variable $-$ velocity $v$

Independent variables $-$ time $t$

Parameters

$,$massm$(kg),$

drag coefficient $c_d(k\frac{g}{m}),$

$,$ gravitational acceleration $g$

Assumed Parameter Values:

$m=68\mathrm{.}1kg,$

$c_d=0\mathrm{.}25k\frac{g}{m},$

$g=9\mathrm{.}81\frac{m}{s}$

Euler's Formula: $v(t_{i+1})=v(t_i)+[g-\frac{c_d}{m}{v}^2(t_i)](t_{i+1}-t_i)$

Analytical solution: $v(t)=\sqrt[\phantom{2}]{\frac{gm}{c_d}}tanh(\sqrt[\phantom{2}]{\frac{g{c}_d}{m}}t)$

MATLAB Simulation:

$,$ Time $($t$)$ range: $0$ to $20$ in a step of $0\mathrm{.}5$

Initial velocity: $v_0=0$

Program Structure: