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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 56 "Maple Worksheet for the
 Solution for Bonus Problem No. 1" }}{PARA 256 "" 0 "" {TEXT 257 68 "A
 Trajectory with Quadratic Drag that is also air-density dependent," }
}{PARA 256 "" 0 "" {TEXT -1 77 "and compared with no dependence on air
 density as well as with no drag at all" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 3 "   " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
107 "    We begin the problem by setting up the entire system of diffe
rential equations that needs to be solved." }}{PARA 0 "" 0 "" {TEXT 
-1 68 "It is best to write them all as first-order differential equati
ons; " }}{PARA 0 "" 0 "" {TEXT -1 35 "therefore, we want an equation f
or " }}{PARA 0 "" 0 "" {TEXT -1 87 "      1) the time-derivative of th
e x-component of the velocity, which we denote by vx," }}{PARA 0 "" 0 
"" {TEXT -1 87 "      2) the time-derivative of the y-component of the
 velocity, which we denote by vy," }}{PARA 0 "" 0 "" {TEXT -1 57 "    \+
  3) the time derivative of x, which should equal vx," }}{PARA 0 "" 0 
"" {TEXT -1 57 "      4) the time derivative of y, which should equal \+
vy," }}{PARA 0 "" 0 "" {TEXT -1 89 "and of course, also the dependence
 of the quadratic drag coefficient, which depends on y." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We are interested in
 three different cases, actually," }}{PARA 0 "" 0 "" {TEXT -1 100 "   \+
a) when the drag coefficient depends on y, i.e., it's some constant mu
ltiplied by exp(-y/lambda);" }}{PARA 0 "" 0 "" {TEXT -1 105 "   b) whe
n the drag coefficient is just the constant, that one gets by setting \+
y=0 in the equation above," }}{PARA 0 "" 0 "" {TEXT -1 87 "   c) and w
hen the drag coefficient is simply zero, as if there were no air resis
tance." }}{PARA 0 "" 0 "" {TEXT -1 105 "We begin, then, by setting up \+
that drag coefficient, where c0 is a constant, depending on the resist
ance " }}{PARA 0 "" 0 "" {TEXT -1 95 "of the air and the mass of the o
bject, and lambda, which I will just call L in the typescript, " }}
{PARA 0 "" 0 "" {TEXT -1 66 " is the atmospheric scale length, given t
o us as 7.600 kilometers." }}{PARA 0 "" 0 "" {TEXT -1 107 "In principl
e the coefficient for the drag should be called c; however, when it is
 inserted into the actual " }}{PARA 0 "" 0 "" {TEXT -1 105 "differenti
al equations that constitute Newton's second Law, that c will be divid
ed by m, so that we need " }}{PARA 0 "" 0 "" {TEXT -1 106 "some new qu
antity, c/m, which I will call w, which is then the constant w0 times \+
the exponential decrease." }}{PARA 0 "" 0 "" {TEXT -1 100 "Since we kn
ow that---when measured in MKS units---the quantity c is given by D*D/
4, we may re-write " }}{PARA 0 "" 0 "" {TEXT -1 108 "the mass m as the
 density, rho, multiplied by the volume, which is (Pi/6)*D*D*D, we can
 write out explicitly" }}{PARA 0 "" 0 "" {TEXT -1 92 "what the constan
t w0 should be that multiplied the exponential decrease with height, n
amely " }}{PARA 0 "" 0 "" {TEXT -1 49 "w0 = (D^2/4)/[rho*(Pi/6)*D^3] =
 (3/2*Pi)/(rho*D)." }}{PARA 0 "" 0 "" {TEXT -1 108 "Since this just mu
ltiplies the square of the velocity to give the acceleration, it must,
 again in MKS units," }}{PARA 0 "" 0 "" {TEXT -1 120 "have dimensions \+
of 1/meter.  Therefore, if I calculate it that way, and then multiply \+
it by the factor (1000 meter/1 km)" }}{PARA 0 "" 0 "" {TEXT -1 91 "I w
ill have the result in kilometers per second, which is the way I want \+
to measure things." }}{PARA 0 "" 0 "" {TEXT -1 118 "This gives us  the
 value for w0.  I want to insert that value as late as possible, so I \+
will first calculate it here, " }}{PARA 0 "" 0 "" {TEXT -1 101 "and ca
ll it w00.  I use the given values for the density and diameter, but a
ll measured in MKS units:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "rho:=7800; Dd:=0.15;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoG\"%
+y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DdG$\"#:!\"#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "w00:=(3/(2*evalf(Pi)))*1000/(rho*Dd
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$w00G$\"+wf*33%!#5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "w:=w0*exp(-y(t)/L);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"wG*&%#w0G\"\"\"-%$expG6#,$*&-%\"yG6#%\"tGF'%\"
LG!\"\"F2F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "wwo:=limit(w
,L=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$wwoG%#w0G" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "The next pair of equations are ju
st the x- and y-components of Newton's Second Law for this situation,
" }}{PARA 0 "" 0 "" {TEXT -1 100 "with all terms moved to the same sid
e of the equation, so that the right-hand side of this equation " }}
{PARA 0 "" 0 "" {TEXT -1 80 "should just be zero, which is what Maple \+
will assume when nothing else is given." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "xdf:=diff(vx(t),t)+w*sqrt(vx(t)^2+vy(t)^2)*vx(t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xdfG,&-%%diffG6$-%#vxG6#%\"tGF,\"\"
\"**%#w0GF--%$expG6#,$*&-%\"yGF+F-%\"LG!\"\"F8F-,&*$)F)\"\"#F-F-*$)-%#
vyGF+F<F-F-#F-F<F)F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "y
df:=diff(vy(t),t)+g+w*sqrt(vx(t)^2+vy(t)^2)*vy(t);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$ydfG,(-%%diffG6$-%#vyG6#%\"tGF,\"\"\"%\"gGF-**%#w0
GF--%$expG6#,$*&-%\"yGF+F-%\"LG!\"\"F9F-,&*$)-%#vxGF+\"\"#F-F-*$)F)F?F
-F-#F-F?F)F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Lastly, we have
 to tell the system how the derivatives of x and y are related to vx a
nd vy" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ddx:=diff(x(t),t)-
vx(t); ddy:=diff(y(t),t)-vy(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$
ddxG,&-%%diffG6$-%\"xG6#%\"tGF,\"\"\"-%#vxGF+!\"\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$ddyG,&-%%diffG6$-%\"yG6#%\"tGF,\"\"\"-%#vyGF+!\"\"
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "That constitutes the equation
s, so we now put them all together:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "syst:=\{xdf,ydf,ddx,ddy\};" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%%systG<&,(-%%diffG6$-%#vyG6#%\"tGF-\"\"\"%\"gGF.**%#w
0GF.-%$expG6#,$*&-%\"yGF,F.%\"LG!\"\"F:F.,&*$)-%#vxGF,\"\"#F.F.*$)F*F@
F.F.#F.F@F*F.F.,&-F(6$F>F-F.**F1F.F2F.F;FCF>F.F.,&-F(6$-%\"xGF,F-F.F>F
:,&-F(6$F7F-F.F*F:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "   The nex
t thing to do is to take the limit as L goes to infinity, on these equ
ations, which will give us the \"second set\"" }}{PARA 0 "" 0 "" 
{TEXT -1 82 "of equations, where the change in air resistance due to a
ltitude is being ignored." }}{PARA 0 "" 0 "" {TEXT -1 99 "This will ma
ke the air resistance somewhat higher at high altitudes than it is the
 equations above." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "consys
t:=map(limit,syst,L=infinity);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(c
onsystG<&,(-%%diffG6$-%#vyG6#%\"tGF-\"\"\"%\"gGF.*(%#w0GF.,&*$)-%#vxGF
,\"\"#F.F.*$)F*F7F.F.#F.F7F*F.F.,&-F(6$-%\"xGF,F-F.F5!\"\",&-F(6$-%\"y
GF,F-F.F*F@,&-F(6$F5F-F.*(F1F.F2F:F5F.F." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 129 "We might as well also now go ahead and create the equati
ons that constitute the case when one totally ignores the air resistan
ce," }}{PARA 0 "" 0 "" {TEXT -1 125 "which we can do by setting w0 to \+
zero.  Of course in that case we could have solved the differential eq
uations and presented " }}{PARA 0 "" 0 "" {TEXT -1 77 "their solutions
 and simply graphed them.  But, we can do it this way as well:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "nosyst:=subs(w0=0,consyst);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'nosystG<&,&-%%diffG6$-%#vyG6#%
\"tGF-\"\"\"%\"gGF.,&-F(6$-%\"xGF,F-F.-%#vxGF,!\"\"-F(6$F5F-,&-F(6$-%
\"yGF,F-F.F*F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "The next comma
nd sets up the initial conditions for this system of first-order ode's
, using the initial speed and initial angle," }}{PARA 0 "" 0 "" {TEXT 
-1 41 "and beginning the motion from the origin:" }{MPLTEXT 1 0 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 64 "init:=[vx(0)=v0*cos(theta0),vy(0)=v0*sin(theta0),x(0)=0,y(0)=0];
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%initG7&/-%#vxG6#\"\"!*&%#v0G\"
\"\"-%$cosG6#%'theta0GF-/-%#vyGF)*&F,F--%$sinGF0F-/-%\"xGF)F*/-%\"yGF)
F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "   Next, we put in a choic
e of parameters for the problem at hand, where I am measuring everythi
ng in kilometers, and seconds." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 63 "g:=9.8/1000; v0:=0.400; theta0:=evalf(Pi)/3; L:=7.600; w0:=w00
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG$\"+++++)*!#7" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#v0G$\"$+%!\"$" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'theta0G$\"+^v>Z5!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"L
G$\"%+w!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#w0G$\"+wf*33%!#5" }}
}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 94 "The command just below tells Maple to load a \"package\" \+
of commands that will allow us to plot " }}{PARA 0 "" 0 "" {TEXT -1 
44 "the solutions of our differential equations." }{MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "
" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined
\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "And now we can begin the so
lution of the three different systems of differential equations:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "Maple has various different ways \+
to do this.  One of the best is just \"dsolve\" and \"odeplot\" which \+
I am using below." }}{PARA 0 "" 0 "" {TEXT -1 128 "I note that the com
mand \"op\" when acting on a list---in brackets---or a set---in (curly
) braces---simply removes those brackets " }}{PARA 0 "" 0 "" {TEXT -1 
109 "or braces, leaving the result as a sequence, so that it can more \+
easily be combined with some other sequence." }}{PARA 0 "" 0 "" {TEXT 
-1 125 "So the use of dsolve creates a Maple procedure to do a numeric
al solution of the specific differential equation with initial " }}
{PARA 0 "" 0 "" {TEXT -1 123 "conditions.  Then odeplot uses that proc
edure to create numerical values over the range that it needs for the \+
plot desired." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 82 "We first do that for the complete problem, with altitude-
dependent air resistance:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
85 "nutri:=dsolve(\{op(syst),op(init)\},[x(t),y(t),vx(t),vy(t)],type=n
umeric,range=0..100):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "ww
ww:=odeplot(nutri,[x(t),y(t)],0..41,refine=2,color=blue,thickness=2):
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Next we do the same thing whe
n the air resistance does not decrease with altitude:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "conn:=dsolve(\{op(consyst),op(init)
\},[x(t),y(t),vx(t),vy(t)],type=numeric,range=0..100):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "wwcon:=odeplot(conn,[x(t),y(t)],0..
40,color=red,thickness=1):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "And
 then yet again the same thing, but now with NO air resistance" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "nonn:=dsolve(\{op(nosyst),op
(init)\},[x(t),y(t),vx(t),vy(t)],type=numeric,range=0..100):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "wwnon:=odeplot(nonn,[x(t),y(
t)],0..71,color=black):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Having
 saved all three of those graphs, we now plot them together." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "display([wwww,wwcon,wwnon],t
itle=`at 60 degrees, initial angle`);" }}{PARA 13 "" 1 "" {GLPLOT2D 
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\"xF)Q\"yF)-%%FONTG6#%(DEFAULTG-%&TITLEG6#%=at~60~degrees,~initial~ang
leG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1
" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Now, let
's change the initial angle to 30 degrees, and look at the results." }
}{PARA 0 "" 0 "" {TEXT -1 103 "Recall that for no air resistance the r
ange is the same for 60 and 30 degrees, since they are the same " }}
{PARA 0 "" 0 "" {TEXT -1 92 "distance on both sides from 45 degrees, a
lthough, of course, the heights are very different." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 20 "theta0:=evalf(Pi)/6;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'theta0G$\"+dx)fB&!#5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "init2:=[vx(0)=v0*cos(theta0),vy(0)=v0*sin(theta0),x(0
)=0,y(0)=0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&init2G7&/-%#vxG6#\"
\"!$\"+:;5kM!#5/-%#vyGF)$\"+++++?F-/-%\"xGF)F*/-%\"yGF)F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "nutri2:=dsolve(\{op(syst),op(init2)
\},[x(t),y(t),vx(t),vy(t)],type=numeric,range=0..100):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "wwww2:=odeplot(nutri2,[x(t),y(t)],0
..26,refine=2,color=blue,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 89 "conn2:=dsolve(\{op(consyst),op(init2)\},[x(t),y(t),vx
(t),vy(t)],type=numeric,range=0..100):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 63 "wwcon2:=odeplot(conn2,[x(t),y(t)],0..26,color=red,thi
ckness=1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "nonn2:=dsolve
(\{op(nosyst),op(init2)\},[x(t),y(t),vx(t),vy(t)],type=numeric,range=0
..100):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "wwnon2:=odeplot(
nonn2,[x(t),y(t)],0..41,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "display
([wwww2,wwcon2,wwnon2],title=`at 30 degrees, initial angle`);" }}
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"Curve 3" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "The two sets of gra
phs appear \"almost\" the same; there is a very slight difference in t
he two graphs for the disparity " }}{PARA 0 "" 0 "" {TEXT -1 36 "betwe
en the red and the blue curves." }}{PARA 0 "" 0 "" {TEXT -1 102 "Howev
er, this similarity is misleading because Maple has changed the vertic
al scale on the two graphs." }}{PARA 0 "" 0 "" {TEXT -1 36 "Let's plot
 all 6 on the same figure:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
48 "display([wwww2,wwcon2,wwnon2,wwww,wwcon,wwnon]);" }}{PARA 13 "" 1 
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0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2
" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 110 "For amusement we can also use the calculations we have m
ade to plot, for instance the horizontal component of " }}{PARA 0 "" 
0 "" {TEXT -1 57 "velocity as a function of time, for the 60 degree la
unch." }}{PARA 0 "" 0 "" {TEXT -1 66 "We see that it slows down, in a \+
fairly nonlinear way, toward zero." }}{PARA 0 "" 0 "" {TEXT -1 107 "It
 really comes eventually to zero; however, looking at a graph somewhat
 further down in this presentation," }}{PARA 0 "" 0 "" {TEXT -1 112 "w
e see that it strikes the ground about t=40, so that the remainder of \+
the graph does not actually happen unless" }}{PARA 0 "" 0 "" {TEXT -1 
55 "there's a high cliff on which all of this is happening." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "odeplot(nutri,[t,vx(t)],0..1
00,refine=2,color=green);" }}{PARA 13 "" 1 "" {GLPLOT2D 539 539 539 
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" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "As a last comment, related \+
to the fact that the horizontal velocity went to zero, is the graph be
low," }}{PARA 0 "" 0 "" {TEXT -1 105 "which shows that x(t) started ou
t nicely, and then is levelling off, never getting beyond 3.5 kilomete
rs." }}{PARA 0 "" 0 "" {TEXT -1 60 "Compare the blue curve above, wher
e we also see that effect." }}{PARA 0 "" 0 "" {TEXT -1 55 "However, ag
ain, it hits the ground again at about t=40." }}}{EXCHG {PARA 0 "> " 
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Q\"xF)-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F7F6" 1 2 0 1 10 0 2 6 1 
4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 84 "Deciding that we should also wonder about the vertic
al direction, the plot below is " }}{PARA 0 "" 0 "" {TEXT -1 89 "the v
ertical velocity, which of course starts out positive, decreases to ze
ro about t=19," }}{PARA 0 "" 0 "" {TEXT -1 83 "and then continues to d
ecrease, overshoots, and starts coming back toward a final, " }}{PARA 
0 "" 0 "" {TEXT -1 36 "terminal velocity in that direction." }}{PARA 
0 "" 0 "" {TEXT -1 84 "Actually in this graph it's not completely clea
r that it's not headed back to zero; " }}{PARA 0 "" 0 "" {TEXT -1 109 
"therefore, let's look at a much larger time, in the next figure, even
 though it doesn't fit with the physics " }}{PARA 0 "" 0 "" {TEXT -1 
56 "of this problem, where it hits the ground at about t=40." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "odeplot(nutri,[t,vy(t)],0..1
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1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 85 "Lastly, let's look at a graph of the vertical location it
self, where we can, finally," }}{PARA 0 "" 0 "" {TEXT -1 48 "see that \+
it hits the ground again at about t=41." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "odeplot(nutri,[t,y(t)],0..50,refine=3,color=blue);" }
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