The Task: $dt=0\mathrm{.}01s$

For each time step case, please have your Python program plot a figure of $v(\frac{m}{s})$ vs$.t(s)$ of the Explicit

Euler Solution based on the optimally designed drag coefficient $\text{'}c\text{'}$ you obtained from the bisection

method. You should have five figures in total for this project once you complete the five different time

steps cases. Also, for each time step case, please list the root prediction evolution by the bisection

method regarding the $c_{-}L,c_{-}U,c_{-}R,$ and the tolerance $E_{-}$a till $E_{-}a\frac{?}{\text{A}\text{B}\text{S}}(f(c))E_{-}a.$ The tolerance $E_{-}ais$ defined $as\frac{?}{A}$

$BS(f(c)).$

Your project report should contain $(i)$ cover page $(also$ list whom are $in$ your programing discussion

group$),(ii)$ problem description, $(iii)$ flow chart, $(iv)$ Python source codes $(if$ you have multiple $.py$ files,

you should upload all $of$ them$),(v)$ results and discussion $(including$ plots, your discussion & analysis

from the result$\frac{s}{e}$rrors comparison$),(vi)$ conclusions. $In$ addition $to$ the project report, a copy $of$ your

python code file $"$

proj$2.py"(and$ other non$-$standard module$\frac{s}{f}$iles created $by$ yourselves$)$ should also $be$

uploaded $to$ CANVAS $so$ I can test$-$run your python code while grading your project report.

In this Project $2,$ your group is asked to design and develop Python program that combines Explicit Euler

Method of the Parachutist problem with your newly learned Bisection Method to find the Root of:

Design Objective Function $=$ Terminal velocity $($predicted by Explicit Euler Method$)-$ Target Terminal

Velocity

Your customized parachute design is required to meet the following specifications:

$m=200\mathrm{.}00kg,g=9\mathrm{.}81\frac{m}{s^2},$ and

Desired Target Terminal Velocity $=5\mathrm{.}50\frac{m}{s}$ with the tolerance less than $+-0\mathrm{.}05\frac{m}{s}($i$.$e$.,E_s=0\mathrm{.}05)$

Your group will need to design the optimal drag coefficient $\text{'}c\text{'}$ so that

Design Objective Function $)=$Terminal Velocity $(c)-$ Target Terminal Velocity $=0$

You will create a bisection method written in a Python Program that will automatically call the codes

modified from your Project $1\text{'}$s Python program$($s$)$ to retrieve the Explicit Euler Method approximated

Terminal Velocity values at different drag coefficient values. And the bisection method program will run

iterations to automatically find the root, which is the optimize drag coefficient. In this project, for your

learning purpose and the full experience of the world of numerical methods, please ignore and do not

use the exact solution in your project $1.$

Your Python program will be called as $"$

proj$2.$py$"$ so you know it is the file for Project $2.$ If you created

multiple modules, you can name the other modules $($rather than the main program $"$

proj$2.$py$")$ any

meaningful filenames you think are appropriate. You will need to put comments in the code and also

add your name and $J$ number in the beginning comments of the code.

Please use the following five different time steps for your numerical design of the drag coefficient $\text{'}c\text{'}$ and

compare and comment about their differences. Your parachute simulations should cover from $0\mathrm{.}0$ s to

$1000\mathrm{.}0$ s$.$

$dt=10\mathrm{.}0s$

$dt=5\mathrm{.}0s$

$dt=1\mathrm{.}0s$