\nopagenumbers
\magnification=1200
\baselineskip=14pt
\def \d{\displaystyle}
\def \h{\hfill}
\def \v{\vfill}
\def \title#1{\line{\h{\bf Math 151  #1}\h}}
\def \namdat#1{\line {Name and Section \vrule height .4pt width2in depth0pt\h#1\par}}
\def \frac #1#2{{\d #1\over \d #2}}
\parindent=0pt
\font \bigbf = cmr10 scaled 1200
\def \i{\item}
\def \vs{\vskip 5pt}
\def \vss{\vskip 10pt}
\def \hs{\hskip 10pt}
\def \c{\circ}
\def \t{\theta}
\def \vc#1{ {\bf\vec#1} }
\def \q{\quad}


\baselineskip = 14pt
\line{\h  Spring 1998}
\line{\h \bigbf Maple Lab, Week 25 \h}
\vss
\line{\h \bigbf Two-Dimensional Motion of a Projectile with Drag \h}
\vss
{\bf Background:}   Consider a point mass moving through the air under the 
influence of gravity and a drag force due to air resistance.  We will take the
drag force to be proportional to the square of the speed and acting in the 
direction opposite to the velocity.  
Applying Newton's second law, the equation of motion in vector form is:
$$ m\frac{d\vc{v}}{dt} = -k\vc{v}|\vc{v}| - m\vc{g} .\leqno(1)$$

Actually, this first-order system of differential equations for the velocity 
vector $\vc{v}(t)$ is a second-order system for the position vector $\vc{r}(t)\,
=\,x(t)\vc{i} + y(t)\vc{j}$.  
In fact, if we write $\vc{v}(t) = v_x(t)\vc{i} + v_y(t)\vc{j}$, 
then we must have $x'(t) = v_x(t),\ y'(t) = v_y(t)$.  In addition,
$ |\vc{v}| = \sqrt{v_x(t)^2 + v_y(t)^2}.\,$  Consequently,  equation (1) gives
rise to the following system of four first-order differential equations:
$$x'(t) = v_x(t),\q y'(t) = v_y(t),\leqno(2)$$
$$v_x'(t) = -\,\frac k m v_x(t)\sqrt{v_x(t)^2 + v_y(t)^2},\q v_y'(t) 
= -\,\frac k m v_y(t)\sqrt{v_x(t)^2 + v_y(t)^2} - g.$$  

We wish to use Maple's {\tt dsolve} command to obtain a numerical solution 
to this problem.    
First, we assume the point mass is at the origin at time $t = 0$ and is 
projected at speed $v_0 = 100$ ft/s at an angle $\alpha = \pi/4$ 
radians above the horizontal.  Also, we take $k/m = 0.0025$, in consistent units.
(Remark: There are examples in the on-line help showing  the use of {\tt dsolve} for
solving systems of equations.  See also Lab 26.)
\vss

{\bf Exercises:}   Use Maple's {\tt dsolve} command with the {\tt numeric} option, 
together with the {\tt plot} command, to produce the following graphs.
[Hint:  Look ahead to Lab 26 (second page) for examples of using the {\tt plot} and 
{\tt display} commands for graphs like these.]

\vss
{\bf 1.}  Plot the trajectory of the solution to the system (2) in the $(x,y)$
plane, for $0\leq t \leq 4.$
\vss
{\bf 2.}  Plot on one coordinate system the above graph together with the 
trajectory for the solution to the same problem without drag. 
\vss
{\bf 3  (extra credit).} Experiment with various values of $v_0$ and $\alpha$, searching
for initial conditions that will make the projectile with drag land ($y = 0$) 
at the same point $x$ where the projectile without drag landed with the original 
initial conditions.
 
\bye




