$*$ The project is done in Microsoft Visual Studio $2022($x$86$ and SDL$2$ used$).$ The photo is Main.cpp to show the desired structure. The requirements for the project are that a maximized window with a title must be created for both video simulations.

$**$Problem:

You run into Galileo outside, and he's going to climb the Leaning Tower of Pisa with two balls.

One ball is heavy, and the other is light, and he says he wants to prove that both balls would reach the

ground at the same time. You tell him, in fact, he doesn't need to work hard in this way, and you will

show him some simulations. You then take him to a cafe for a drink, and then you open your laptop and

run your program.

Two balls of different colors appear on your screen, and their initial positions are at the same height. For

example, one ball's initial position is at $(-5,25)$ and the other is at $(5,25).$ Their weights differ by a

hundred times. For example, the heavy ball is $10$ kg$,$ while the light ball is only $0\mathrm{.}1$ kg$.$ They're of the

same size, because you're going to assume they experience the same drag force later.

You make two simulations for Galileo. The first simulation ignores the resistance of the air, and it

demonstrates that the two balls of different masses reach the ground at the same time. The second

simulation considers the resistance of the air. For the sake of simplicity, you assume they experience the

same drag, since they are of the same size, shape and smoothness, the only difference is the mass. In the

second simulation, you apply a small drag force to both balls. The Simulation shows that the heavier ball

reaches the ground first. It$$s better to set the bouncing ratio to zero so when a ball hits the ground it

stops. The drag force for the two balls is set the same, and you decide how strong it is$.$

$**$Application$.$h

#ifndef APPLICATION$\_$H

#define APPLICATION$\_$H

#include $"./$Graphics$.$h$"$

#include $"./$Physics$/$Particle$.$h$"$

#include

class Application $\{$

private:

bool running $=$ false;

std::vector particles;

Vec$2$ pushForce $=$ Vec$2(0,0)$;

Vec$2$ mouseCursor $=$ Vec$2(0,0)$;

bool leftMouseButtonDown $=$ false;

public:

Application$()=$ default;

~Application$()=$ default;

bool isRunning$()$;

void setup$()$;

void input$()$;

void update$()$;

void render$()$;

void destroy$()$;

$\}$;

#endif

$***$Code present in completed applications:

#include "Application.h$"$

#include $"./$Physics$/$Constants$.$h$"$

#include $"./$Physics$/$Force$.$h$"$

#include

bool Application::isRunning$()\{$

return running;

$\}$

void Application::setup$()\{$

running $=$ Graphics::openWindow$("$A single particle moving within a rectangle"$)$;$//$openWindow$()$;

$//------$ set up the position of the coordinate system origin in the window $------$

Graphics::setOrigin$($Graphics::windowWidth $/2,$ Graphics::windowHeight $/2)$;

Particle$*$ aDot $=$ new Particle$(0,0,10,0\mathrm{.}5)$;

particles.push$\_$back$($aDot$)$;

$\}$

void Application::input$()\{$

SDL$\_$Event event;

while $($SDL$\_$PollEvent$($&event$))\{$

switch $($event$.$type$)\{$

case SDL$\_$QUIT:

running $=$ false;

break;

void Application::update$()\{$

static int timePreviousFrame;

int timeToWait $=$ MILLISECS$\_$PER$\_$FRAME $-($SDL$\_$GetTicks$()-$ timePreviousFrame$)$;

if $($timeToWait $>0)$

SDL$\_$Delay$($timeToWait$)$;

float deltaTime $=($SDL$\_$GetTicks$()-$ timePreviousFrame$)/1000\mathrm{.}0$f;

if $($deltaTime $>0\mathrm{.}016)$

deltaTime $=0\mathrm{.}016$;

timePreviousFrame $=$ SDL$\_$GetTicks$()$;

for $($auto particle : particles$)\{$

particle$->$addForce$($pushForce$)$;

Vec$2$ friction $=$ Force::generateFrictionForce$(*$particle$,40)$;

if$($particle$->$velocity.magnitude$()>0\mathrm{.}001)$

particle$->$addForce$($friction$)$;

$\}$

for $($auto particle : particles$)\{$

particle$->$integrate$($deltaTime$)$; $//$ Integrate the acceleration and velocity to estimate the new position

$\}$

for $($auto particle : particles$)\{$

if $($Graphics::xPosInWindow$($particle$->$position.x $-$ particle$->$radius

particle$->$position.x $=$ Graphics::xPosInCoordinate$(0)+$ particle$->$radius;

particle$->$velocity.x $*=-$ratio;

particle$->$velocity.y $*=$ ratio;

$\}$

else if $($Graphics::xPosInWindow$($particle$->$position.x $+$ particle$->$radius$)>=$ Graphics::getWindowWidth$())\{$

particle$->$position.x $=$ Graphics::xPosInCoordinate$($Graphics::getWindowWidth$())-$ particle$->$radius;

particle$->$velocity.x $*=-$ratio;

particle$->$velocity.y $*=$ ratio;

$\}$

if $($Graphics::yPosInWindow$($particle$->$position.y $-$ particle$->$radius$)>=$ Graphics::getWindowHeight$())\{$

particle$->$position.y $=$ Graphics::yPosInCoordinate$($Graphics::getWindowHeight$())+$ particle$->$radius;

particle$->$velocity.x $*=$ ratio;

particle$->$velocity.y $*=-$ratio;

$\}$

else if $($Graphics::yPosInWindow$($particle$->$position.y $+$ particle$->$radius

particle$->$position.y $=$ Graphics::yPosInCoordinate$(0)-$ particle$->$radius;

particle$->$velocity.x $*=$ ratio;

particle$->$velocity.y $*=-$ratio;

$\}$

$\}$

$\}$

void Application::render$()\{$

x

x

Graphics::renderFrame$()$;

$\}$

void Application::destroy$()\{$

for $($auto particle: particles$)\{$

delete particle;

$\}$

Graphics::CloseWindow$()$;

$\}$