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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Andrew Sedacki" }}{PARA 0 
"" 0 "" {TEXT -1 20 "Project Two MCS 320 " }}{PARA 0 "" 0 "" {TEXT -1 
20 "Monday 29 March 2004" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 8 "restart;" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 30 "0
. A simple model of billiards" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 241 "
The goal of this project is to gain experience with functions and the \+
plotting capabilities in Maple to define a simplistic model of billiar
ds.  Despite the simplicity of our model, we will see trajectories nor
mally not seen on a pool table." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 
40 "1. Defining and visualizing trajectories" }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 61 "Imagine a square pool table.  The length of each side i
s one." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
10 "a:=1:b:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "f0:=plott
ools[line]([0,0],[a,0],color=green,thickness=3):" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 57 "f1:=plottools[line]([0,0],[0,b],color=green,t
hickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "f2:=plottoo
ls[line]([a,0],[a,b],color=green,thickness=3):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 57 "f3:=plottools[line]([0,b],[a,b],color=green,th
ickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "pool:=f0,f1,
f2,f3:  # this stores the plots for later rendering" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 156 "A billiard ball moving on the pool table start
s at some position (x0,y0) and moves with constant speed (vx,vy).  We \+
take random numbers for these constants." }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "x0,y0,vx,vy:=stats[random,un
iform[0,a]](4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6&%#x0G%#y0G%#vxG%
#vyG6&$\"+\"p'>uU!#5$\"+Lp56KF,$\"+P2LOMF,$\"+O9cUZF," }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 35 "Then the ball moves along the line " }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "xt:=t->
x0+vx*t: yt:=t->y0+vy*t:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "To co
mpute the position of the ball on the pool table, we need the followin
g functions:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 36 "r_mod_one:=x->[trunc(x),x-trunc(x)]:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 53 "position:=nr->piecewise(nr[1] mod 2=0,nr[2],1-nr
[2]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "xp:=position@r_mod
_one@xt: yp:=position@r_mod_one@yt:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 47 "Plotting the trajectory is now straightforward:" }{MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "path:=plot([xp*a,yp*
b,0..10]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "plots[display
](pool,path,scaling=constrained,axes=none);" }}{PARA 13 "" 1 "" 
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\"\\:F\\r$\"*W$*3i*F\\r7$$\"*%f\\i8F\\r$\"*ryKO*F\\r-F.6&F0$\"#5!\"\"F
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45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "And we can make an animati
on:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "N
:=100: # number of frames" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "t:=0.0
: dt:=0.05:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "path:=[]: # list of \+
plots" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for i from 1 to N do" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "  pt:=[xp(t)*a,yp(t)*b]:" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 43 "  ball:=plottools[disk](pt,0.01,color=red
):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "  path:=[op(path),plots[displ
ay](pool,ball,scaling=constrained,axes=none)]:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "  t:=t+dt:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end d
o:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "To launch the animation, e
xecute the following command, click on the picture and then use the to
olbar." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
57 "plots[display](path,insequence=true,scaling=constrained);" }}
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "Square pool tables are very rare.
  Therefore, the first assignment is to extend this model to any recta
ngular pool table." }{MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 14 "Assignment One" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Mod
ify the model to work with any rectangular pool table of dimensions a \+
and b." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 339 "Cre
ate a function \"pool\" which takes as arguments the dimensions a and \+
b, and which returns the plot of a rectangular pool table with sides o
f length a and b.  For example, pool(3,2) returns the plot of a 3-by-2
 rectangular pool table.  Give the definition of this function and sho
w the plot command (with its output) to test the function." }{MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 306 "Use the functions r_m
od_one and position from above to define two indexed functions rxp and
 ryp called as rxp[a](t) and ryp[b](t), which give the position of a b
illiard ball at time t on a pool table of dimensions a and b.  As befo
re, use the composition with the functions xt and yt to define rxp and
 ryp." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "pool:=proc(a,b)" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "f0:=plottools[line]([0,0],[a,0],co
lor=green,thickness=3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "f1:=plot
tools[line]([0,0],[0,b],color=green,thickness=3):" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 57 "f2:=plottools[line]([a,0],[a,b],color=green,thickne
ss=3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "f3:=plottools[line]([0,b]
,[a,b],color=green,thickness=3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 
"pool:=f0,f1,f2,f3:  # this stores the plots for later rendering" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "x0,y0,vx,vy:=stats[random,uniform[0
,a]](4):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "xt:=t->x0+vx*t: yt:=t->
y0+vy*t:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "r_mod_one:=x->[trunc(x)
,x-trunc(x)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "position:=nr->piec
ewise(nr[1] mod 2=0,nr[2],1-nr[2]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
53 "xp:=position@r_mod_one@xt: yp:=position@r_mod_one@yt:" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 30 "path:=plot([xp*a,yp*b,0..10]):" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 56 "plots[display](pool,path,scaling=constrained,
axes=none);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end proc:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "pool(3,2);" }}{PARA 13 "" 1 "" 
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AXESSTYLEG6#%%NONEG-%%VIEWG6$%(DEFAULTGFh[`l" 1 2 0 1 10 0 2 9 1 1 1 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" "Curve 6" "Curve 7" }}}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 45 "3. Slowing down the speed of the trajectories" }}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 112 "In this current model, the ball moves at const
ant speed, without ever slowing down.  This is not very realistic." }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "Instead of t
he vx*t and vy*t terms in the definitions of xt and yt, we better take
 an exponentially decaying function." }{MPLTEXT 1 0 0 "" }}}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 16 "Assignment Three" }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 104 "Change the function xt and yt taking vx and vy to be 1
0, so that at t = 100, the speed drops below 0.01." }{MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Create an animation with suff
iciently many frames to see the effect." }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "a:=1:b:=1:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 57 "f0:=plottools[line]([0,0],[a,0],color=green,thic
kness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "f1:=plottools[
line]([0,0],[0,b],color=green,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 57 "f2:=plottools[line]([a,0],[a,b],color=green,thickn
ess=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "f3:=plottools[li
ne]([0,b],[a,b],color=green,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 63 "pool:=f0,f1,f2,f3:  # this stores the plots for lat
er rendering" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 156 "A billiard ball \+
moving on the pool table starts at some position (x0,y0) and moves wit
h constant speed (vx,vy).  We take random numbers for these constants.
" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "x0,y
0,vx,vy:=stats[random,uniform[0,a]](4);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>6&%#x0G%#y0G%#vxG%#vyG6&$\"+!>(e%e&!#5$\"+0$QvY(F,$\"+3AA1K!#6$
\"+=7uHsF," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Then the ball moves
 along the line " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "xt:=t->x0+vx^t: yt:=t->y0+vy^t:" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 87 "To compute the position of the ball on the pool ta
ble, we need the following functions:" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "r_mod_one:=x->[trunc(x),x-trunc(x)]
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "position:=nr->piecewis
e(nr[1] mod 2=0,nr[2],1-nr[2]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 53 "xp:=position@r_mod_one@xt: yp:=position@r_mod_one@yt:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Plotting the trajectory is now str
aightforward:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "path:=plot([xp*a,yp*b,0..10]):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "
N:=100: # number of frames" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "t:=0.
0: dt:=0.05:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "path:=[]: # list of
 plots" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for i from 1 to N do" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "  pt:=[xp(t)*a,yp(t)*b]:" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 43 "  ball:=plottools[disk](pt,0.01,color=red
):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "  path:=[op(path),plots[displ
ay](pool,ball,scaling=constrained,axes=none)]:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "  t:=t+dt:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end d
o:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "To launch the animation, e
xecute the following command, click on the picture and then use the to
olbar." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
57 "plots[display](path,insequence=true,scaling=constrained);" }}
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::::::::::::::::::::::::::::::::JycynYxiwmxsyyip::Zvmy;Zg=::VYjxkyKyCy
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::::::::::::::::::::::::::::::::JycynYxiwmxsyyip::Zvmy;Zg=::VYjxkyKyCy
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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 46 "4. The deadline is Monday 29 Marc
h 2004 at 2PM" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 368 "The solutions to
 the project will be collected at the beginning of our class meeting o
n Monday 29 March at 2PM.  If you cannot come to class that day, then \+
you must arrange to hand in your solution before the deadline.  Otherw
ise, your solution will be discounted with 10 points if it is turned i
n on the same day before 5PM, and will no longer be accepted afterward
s." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 176 "The sol
ution consists of the print out of one single worksheet, organized wit
h sections according to the two assignments.  Also write proper docume
ntation for the calculations." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 59 "To avoid any misunderstanding, this project must b
e solved " }{TEXT 256 12 "individually" }{TEXT -1 158 ".  Under no cir
cumstances is it allowed to copy or to collaborate.  Regardless of who
 copied from whom, all caught in the act of plagiarism will be penaliz
ed." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "If you \+
have questions, comments, or difficulties, feel free to come to my off
ice for help." }}}}}{MARK "6 4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 
1 }{PAGENUMBERS 0 1 2 33 1 1 }