p3_parametic_projectile.tex

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\documentclass
[justified,nohyper]
{tufte-handout}

\usepackage{amsmath}

\usepackage{booktabs}
\usepackage{graphicx}
\usepackage{kmath,kerkis} % The order of the packages matters; kmath changes the default text font
\usepackage[T1]{fontenc}

\usepackage{enumitem}


% USEFUL SHORTCUTS FOR MATH
\newcommand{\ds}{\displaystyle}

\newcommand{\dt}[1]{\dfrac{d#1}{dt}}

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\begin{document}
\begin{fullwidth}
    \mbox{\LARGE PreCalculus BC: Project Three - \today }\hfill
\end{fullwidth}
\section*{Introduction}

As we have seen, parametric equations can be powerful tools in describing
a variety of curves in two-dimensional space. One example of this power
is in describing the path of a projectile. At any moment
in time, the position of the projectile, $P$, is given by
$$
    P: \begin{cases}
            x &= f(t) \\
            y &= g(t)
        \end{cases}
$$
where $f$ and $g$ are functions of $t$.

There are two parts to this project. In the first part, you will develop
parametric equations that describe the flight of a simple cannonball that
is launched from some angle $\theta$ with respect to the horizontal.
In the second part, you will develop
parametric equations that describe the flight of the same cannonball which is
launched under the same condtions, but with a bounce off a wall that is located
a distance of $d$ meters away from the launch site.

\section*{Part One -- The Cannonball}

Modeling motion is one of the most important ideas in both classical and modern
physics. Much of Isaac Newton's work dealt with creating a mathematical model
for how objects move and interact -- this was the main reason of his invention
of the Calculus. Albert Einstein developed his Special Theory of Relativity in the
early 1900s to refine Newton's laws of motion.

We will use coordinate geometry to model the motion of a projectile, such
as a cannonball fired upward into the air. Suppose we fire a projectile
into the air from ground level, with an intial speed of $v_0$ meters per second
and an angle $\theta$, measured in radians,
upward from the ground. If there were no gravity and no air resistance, the projectile
would just keep moving indefinitely at the same speed and in the same direction.
Since distance equals speed times time, the projectile would travel a distance
of $v_0t$, so its position, $P$, at time $t$ would be given by the following
parametric equations.

$$
    P: \begin{cases}
            x &= (v_0 \cos \theta)t \\
            y &= (v_0 \sin \theta)t
        \end{cases}
$$

But, of course, we know that gravity will pull the projectile back to ground
level. Your task in part one is to modify the parametric equation given above
to account for the effects of gravity. In this part, please use $g=-9.8$ meters
per second per second as the gravitational constant. Be sure to completely
describe the development of your new model and include an example graph
showing the position over time of the projectile. This will require that you
plot a parametric equation in SAGE, and we will do this in class.

\section*{Part Two -- The Bounce} 

After launching the cannonball at an angle of $\theta$ with respect to the ground,
it will collide with a wall. You will need to consider a few things.
First, depending on $\theta$, the
cannonball might not even make it to the wall that is a distance of $d$ meters
away. Second, you can assume that the wall is infinitely tall, which means
you would never be able to fire the cannonball over the wall. Third, you can
assume the collision is purely elastic -- which means no energy is lost
in the impact.

Your goal will be to determine a second parametric equation that describes the
flight of the cannonball after the collision. The final result will be two
parametric equations. One for before the collision and one for after the collision.
You must also supply the $t$-values for each of these two parametric equations,
which will indicate how long the cannonball is in flight.

\section*{What I will be looking for in your report}

\begin{itemize}
    \item Have you justified and explained the derivation of the parametric
    equation in Part 1?
    \item Do you have the appropriate graph of the parametric equation in Part 1
    and is it contained in a figure environment with a caption and referenced
    in the text of your report?
    \item Have you justified and explained the derivation of the parametric
    equation in Part 2?
    \item Do you have the appropriate graph of the parametric equation in Part 2
    and is it contained in a figure environment with a caption and referenced
    in the text of your report? This may include the results that we will discuss
    in class about how to handle elastic collisions.
    \item Do you have the correct values for $t$ in both parametric equations,
    representing what happens before and after the collision.
    \item Have you considered at least one potential extension of the ideas
    used in this project and described their potential usefulness.
    \item Does your report use equation formatting properly? All variables
    and equations should be correctly typeset.
    \item There should be no spacing mistakes. For example, in this
    sentence I have not placed a space after the period.This is bad.
\end{itemize}

\end{document}