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\begin{document}

\input{213-Lecture-Macros}		%% 	macros, settings, et.c


\title{Math 213 - Velocity and Acceleration}			%%	title page
\author{Peter A. Perry}
\institute{University of Kentucky}
\date{September 20, 2019}
\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	
%	Due Date Reminders
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Reminders}

\begin{enumerate}
\item	You will receive graded exams on Tuesday in recitation - please return exams with any grading questions by the end of your Tuesday recitation
\item	Homework A6 on 13.3-13.4 is due on \textbf{Monday}
\item	Homework B1 on 14.1 is due on \textbf{Wednesday}
\item	Homework B2 on 14.3 is due on \textbf{Friday}
\end{enumerate}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%	Unit Outline
%
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\begin{frame}
\frametitle{Unit II: Functions of Several Variables}
\begin{itemize}
\begingroup
\color{red}
\item[{13.3-4}] 	Lecture 11: Velocity and Acceleration
\endgroup

\medskip

\item[{14.1}] Lecture 12: Functions of Several Variables
\item[{14.3}] Lecture 13: Partial Derivatives
\item[{14.4}] Lecture 14: Linear Approximation
\item[{14.5}] Lecture 15: Chain Rule, Implicit Differentiation
\item[{14.6}] Lecture 16: Directional Derivatives and the Gradient

\medskip

\item[{14.7}] Lecture 17: Maximum and Minimum Values, I
\item[{14.7}] Lecture 18: Maximum and Minimum Values, II
\item[{14.8}] Lecture 19: Lagrange Multipliers

\medskip

\item[{15.1}]	Double Integrals
\item[{15.2}]	Double Integrals over General Regions
\item[{}]			Exam II Review
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%	Learning Goals
%
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\begin{frame}
\frametitle{Learning Goals}
\begin{enumerate}
\item	Know how to compute velocity and acceleration
\item	Know how to solve projectile problems
\item	Understand how to compute arc length
\end{enumerate}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%		New material begins here...
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\section{Velocity and Acceleration}

\begin{frame}
\frametitle{Velocity and Acceleration}
\footnotesize
If $\bfr(t)$ is the space curve of a moving body and if $t$ is time:
\begin{itemize}
\item	$\bfr'(t)$ is $\bfv(t)$, the \emph{velocity} of the moving body
\item	$|\bfr'(t)|$ is the \emph{speed} of the moving body
\item	$\bfr''(t)$ is $\bfa(t)$, the \emph{acceleration} of the moving body
\end{itemize}

\medskip

\hrulefill

\medskip

\begin{enumerate}
\item		(Projectile motion) Suppose that $\bfr(t) = \langle 32t, 32t-16t^2 \rangle$. Find the velocity and acceleration
\item		(Circular motion) Suppose that $\bfr(t) = \langle R \cos (2\pi t/T), R \sin (2\pi t/T) \rangle$. Find the velocity and acceleration.
\end{enumerate}

\end{frame}

\begin{frame}
\frametitle{Velocity and Acceleration}


\begin{minipage}{0.4\textwidth}
\begin{tikzpicture}[scale=0.4]
\draw[thick,->,>=stealth]	(0,0)	--	(0,2);
\draw[thick,->,>=stealth]	(0,0)	--	(7,0);
\draw[red]	plot[domain=0:2,smooth,variable=\t]		({0.1*32*\t},{0.1*(32*\t-16*\t*\t)});
\end{tikzpicture}
\end{minipage}
\qquad
\begin{minipage}{0.5\textwidth}
\footnotesize
$\bfr(t) = \langle 32t, 32t-16t^2 \rangle$.

\bigskip

What's the projectile's acceleration?

\smallskip

When does the projectile hit the ground?

\smallskip

What is its speed when it hits?

\smallskip

How far does it go?

\smallskip

What is its maximum height?
\end{minipage}

\bigskip

\hrulefill

\bigskip

\begin{minipage}{0.4\textwidth}
\begin{tikzpicture}[scale=0.8]
\draw[thick,->, >=stealth]		(-2,0)	--(2,0);
\draw[thick,->,>=stealth]		(0,-2)	--(0,2);
\draw[red]	plot[domain=0:360,samples=50,smooth]		({1.5*cos(\x)}, {1.5*sin(\x)});
%%
\draw[red, thick,->,>=stealth]	(1.06,1.06)		--	++(-0.5,0.5);
\draw[cyan,thick,->,>=stealth] 	(1.06,1.06)		--	++(-0.5,-0.5);
%%
\end{tikzpicture}
\end{minipage}
\qquad
\begin{minipage}{0.5\textwidth}
\footnotesize
$\bfr(t) = \langle R \cos (2\pi t/T), R \sin(2\pi t/T) \rangle$

\bigskip

How long does one orbit take?

\smallskip

Which way does the velocity vector point?

\smallskip

Which way does the acceleration vector point?
\end{minipage}
\end{frame}

\section{Arc Length}

\begin{frame}
\frametitle{Math 114 Reminder}
\footnotesize
In Math 114, we defined the arc length of a parameterized curve
$$ x = f(t), \quad y = g(t), \quad a \leq t \leq b $$
as
$$ L = \int_a^b \sqrt{f'(t)^2 + g'(t)^2} \, dt. $$

\bigskip

We can now recognize arc length as \emph{the integral of speed}: if
$$ \bfr(t) = f(t) \ii + g(t) \jj $$
then the velocity along the curve is
$$ \bfr'(t) = f'(t) \ii + g'(t) \jj $$
and the speed is
$$ | \bfr'(t) | = \sqrt{f'(t)^2 + g'(t)^2} $$


\end{frame}

\begin{frame}
\frametitle{Arc Length}
\footnotesize
For a space curve $\bfr(t) = x(t) \ii + y(t) \jj + z(t) \kk$, the arc length of the curve between $t=a$ and $t=b$ is:
\begin{align*}
L	&=	\int_a^b \left| \bfr'(t) \right| \, dt \\
	&=	\int_a^b \sqrt{x'(t)^2 + y'(t)^2 + z'(t)^2} \, dt 
\end{align*}

\medskip

\hrulefill

\bigskip

Find the arc length of the curve 
$$ \bfr(t) =  cos(t) \ii + \sin(t) \jj + ln (\cos t) \kk $$
between $t=0$ and $t=\pi/4$.
\end{frame}

\begin{frame}
\frametitle{Interlude - Newton's Laws of Motion}
\footnotesize
\fcolorbox{cyan}{white}
{
\begin{minipage}{0.85\textwidth}
	\begin{enumerate}
	\item		A body will remain at rest or in motion in a straight line unless acted on by an external force.
	\item		The applied force $\bfF$ is equal to the change of momentum $m\bfv$ per unit time
	\item		For every action there is an equal and opposite reaction
	\end{enumerate}
\end{minipage}
}
\end{frame}

\section{Projectile Motion}

\begin{frame}
\frametitle{Projectile Motion}
\footnotesize
For constant mass,  Newton's second law implies
$$ \bfF  = m \bf a $$
(Warning: Do not use for rockets!)

\bigskip

At the surface of the earth, a mass $m$ is subject to a gravitational force $-mg \kk$

\bigskip

From Newton's second law we then get $ m \bfa = - mg \kk $ or
$$\bfr''(t) = \bfa = -g \kk $$
where $g=32\text{ ft/sec}^2 = 9.8 \text{ m/sec}^2$.
\bigskip

If we know the \emph{initial conditions} for a projectile (its position and velocity at time zero), 
we can integrate this equation to find the motion of the projectile.

\end{frame}

\begin{frame}
\frametitle{Projectile Motion  - Metric Units}
\footnotesize
A ball is thrown at an angle of $45^\circ$ to the ground. If the ball lands $90$ m away, what was the initial speed of the ball?

\bigskip 

\begin{align*}
\bfr''(t) 	&=	-9.8 \kk\\
\bfr'(0)	&=	\bfv(0)	=	v_0 \cos(45^\circ) \ii + v_0 \sin(45^\circ) \kk \\
\bfr(0)	&=	0 \ii + 0 \kk
\end{align*}

\bigskip

Now integrate:
\begin{align*}
\bfv(t) 	&=	\bfv(0)	+ \int_0^t \bfa(s) \, ds \\
			&=	v_0 (\sqrt{2}/2) \ii + \left( v_0 (\sqrt{2}/2)  - 9.8 t \right) \kk \\
\bfr(t) 	&=	\bfr(0) + (v_0  (\sqrt{2}/2)t ) \ii +\left( v_0 (\sqrt{2}/2)t - (9.8/2) t^2  \right)  \kk 
\end{align*}

\bigskip	

Now what?

\end{frame}

\begin{frame}
\frametitle{More Fun with Projectile Motion - English Units}
\footnotesize
A rifle is fired with angle of elevation $36^\circ$. What is the muzzle speed if the maximum height of the bullet is $1600$ ft?

\vfill


\end{frame}

\begin{frame}
\frametitle{Yet More Fun with Projectile Motion}
\footnotesize
A batter hits a baseball $3$ft above the ground toward the center field fence, which is $10$ ft high and $400$ ft from home plate. The ball leaves the bat with speed $115$ ft/sec at an angle of $50^\circ$ above the horizontal. It is a home run? (that is, does the ball clear the fence?)

\vfill

\end{frame}

\begin{frame}
\frametitle{Projectile Motion - Some Takeways}

\footnotesize

Given the position function $$\bfr(t) = x(t) \ii + z(t) \kk$$ for a projectile,
how do you determine$\ldots$

\medskip

\begin{itemize}
\item		The maximum height of the projectile?  

			(At what time $t$ does this occur?)
			
\smallskip
\item	 	The range of the projectile?  

			(At what time $t$ does the projectile hit the ground?)
			
\smallskip

\item		The speed of the projectile at impact?
\end{itemize}

\vfill

\end{frame}

\section{Summary}

\begin{frame}
\frametitle{Summary}

We discussed:

\medskip

\begin{itemize}
\item	How to find the velocity, speed, and acceleration from the vector function 
$\bfr(t)$ that describes the motion of a particle in space
\item	How to compute arc length by integrating the speed
\item	How to solve projectile problems
\end{itemize}
\end{frame}





\end{document}