From 83fff87795006193899e4db6e4eee156b759a1da Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Mon, 27 Nov 2017 15:26:42 -0500 Subject: [PATCH] added HW6 mini project competition --- .../18_initial_value_ode.ipynb | 97 +++++++------- 19_nonlinear_ivp/19_nonlinear_ivp.ipynb | 118 ++++++++++++++---- HW6/README.md | 83 +++++++++++- HW6/equations/eq1.png | Bin 0 -> 4754 bytes HW6/equations/eq1.tex | 4 + HW6/equations/eq2.png | Bin 0 -> 4453 bytes HW6/equations/eq2.tex | 4 + HW6/equations/eq3.png | Bin 0 -> 3421 bytes HW6/equations/eq3.tex | 1 + HW6/equations/eq4.png | Bin 0 -> 4301 bytes HW6/equations/eq4.tex | 4 + HW6/equations/eq5.png | Bin 0 -> 4511 bytes HW6/equations/eq5.tex | 4 + HW6/equations/eq6.png | Bin 0 -> 5986 bytes HW6/equations/eq6.tex | 9 ++ HW6/equations/eq7.png | Bin 0 -> 4140 bytes HW6/equations/eq7.tex | 2 + HW6/equations/eq8.png | Bin 0 -> 7104 bytes HW6/equations/eq8.tex | 8 ++ 19 files changed, 252 insertions(+), 82 deletions(-) create mode 100644 HW6/equations/eq1.png create mode 100644 HW6/equations/eq1.tex create mode 100644 HW6/equations/eq2.png create mode 100644 HW6/equations/eq2.tex create mode 100644 HW6/equations/eq3.png create mode 100644 HW6/equations/eq3.tex create mode 100644 HW6/equations/eq4.png create mode 100644 HW6/equations/eq4.tex create mode 100644 HW6/equations/eq5.png create mode 100644 HW6/equations/eq5.tex create mode 100644 HW6/equations/eq6.png create mode 100644 HW6/equations/eq6.tex create mode 100644 HW6/equations/eq7.png create mode 100644 HW6/equations/eq7.tex create mode 100644 HW6/equations/eq8.png create mode 100644 HW6/equations/eq8.tex diff --git a/18_initial_value_ode/18_initial_value_ode.ipynb b/18_initial_value_ode/18_initial_value_ode.ipynb index 110f785..24e1605 100644 --- a/18_initial_value_ode/18_initial_value_ode.ipynb +++ b/18_initial_value_ode/18_initial_value_ode.ipynb @@ -84,7 +84,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 11, "metadata": { "collapsed": false, "slideshow": { @@ -117,7 +117,7 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 18, "metadata": { "collapsed": false, "slideshow": { @@ -132,14 +132,14 @@ "time (s)| error (m/s)\n", "-------------------------\n", " 0.0 | 0.00\n", - " 2.3 | 0.46\n", - " 4.0 | 0.95\n", - " 6.3 | 1.03\n", - " 8.0 | 0.80\n", - " 10.3 | 0.46\n", - " 12.0 | 0.28\n", + " 6.7 | 23.01\n", + " 6.7 | 23.01\n", + " 13.3 | 36.09\n", + " 13.3 | 36.09\n", + " 20.0 | 24.90\n", + " 20.0 | 24.90\n", "\n", - "O(h^2)=0.33\n" + "O(h^2)=44.44\n" ] }, { @@ -195,63 +195,68 @@ "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t10\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t20\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t30\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t40\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t50\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t0\n", + "\t\t\n", + "\t\t60\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t2\n", + "\t\t\n", + "\t\t70\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t4\n", + "\t\t\n", + "\t\t80\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t6\n", + "\t\t\n", + "\t\t0\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t8\n", + "\t\t\n", + "\t\t5\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t10\n", "\t\n", "\n", "\n", + "\t\t\n", + "\t\t15\n", + "\t\n", + "\n", + "\n", "\t\t\n", - "\t\t12\n", + "\t\t20\n", "\t\n", "\n", "\n", @@ -274,33 +279,15 @@ "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_2a\n", "\n", - "\t\t \n", + "\t\t \n", "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", + "\t\n", + "\t\n", + "\t\n", "\n", "\t\n", "\n", @@ -323,7 +310,7 @@ } ], "source": [ - "[v_an,v_t,t]=freefall(22,12);\n", + "[v_an,v_t,t]=freefall(4,20);\n", "fprintf('\\nO(h^2)=%1.2f',min(diff(t).^2))" ] }, @@ -372,7 +359,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 19, "metadata": { "collapsed": false, "slideshow": { @@ -416,7 +403,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 20, "metadata": { "collapsed": false, "slideshow": { @@ -466,7 +453,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 21, "metadata": { "collapsed": false, "slideshow": { @@ -500,7 +487,7 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 22, "metadata": { "collapsed": false, "slideshow": { @@ -537,7 +524,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 23, "metadata": { "collapsed": false, "slideshow": { diff --git a/19_nonlinear_ivp/19_nonlinear_ivp.ipynb b/19_nonlinear_ivp/19_nonlinear_ivp.ipynb index b820bd6..5cc9272 100644 --- a/19_nonlinear_ivp/19_nonlinear_ivp.ipynb +++ b/19_nonlinear_ivp/19_nonlinear_ivp.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 1, + "execution_count": null, "metadata": { "collapsed": true, "slideshow": { @@ -16,7 +16,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": null, "metadata": { "collapsed": true, "slideshow": { @@ -30,7 +30,7 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": null, "metadata": { "collapsed": false, "slideshow": { @@ -51,8 +51,17 @@ }, "source": [ "# Initial Value Problems (continued)\n", - "*ch. 23 - Adaptive methods*\n", - "\n", + "*ch. 23 - Adaptive methods*" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ "## Predator-Prey Models (Chaos Theory)" ] }, @@ -77,7 +86,11 @@ }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, "source": [ "```matlab\n", "function yp=predprey(t,y,a,b,c,d)\n", @@ -93,28 +106,16 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": null, "metadata": { "collapsed": false, "slideshow": { "slide_type": "fragment" } }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "ans =\n", - "\n", - " 12.0000\n", - " 5.2000\n", - "\n" - ] - } - ], + "outputs": [], "source": [ - "predprey(1,[20 1],a,b,c,d)" + "predprey(1,[20 1],1,1,1,1)" ] }, { @@ -515,6 +516,73 @@ "ylabel('thousands of predators')" ] }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "# Fourth Order Runge-Kutte integration\n", + "\n", + "```matlab\n", + "function [tp,yp] = rk4sys(dydt,tspan,y0,h,varargin)\n", + "% rk4sys: fourth-order Runge-Kutta for a system of ODEs\n", + "% [t,y] = rk4sys(dydt,tspan,y0,h,p1,p2,...): integrates\n", + "% a system of ODEs with fourth-order RK method\n", + "% input:\n", + "% dydt = name of the M-file that evaluates the ODEs\n", + "% tspan = [ti, tf]; initial and final times with output\n", + "% generated at interval of h, or\n", + "% = [t0 t1 ... tf]; specific times where solution output\n", + "% y0 = initial values of dependent variables\n", + "% h = step size\n", + "% p1,p2,... = additional parameters used by dydt\n", + "% output:\n", + "% tp = vector of independent variable\n", + "% yp = vector of solution for dependent variables\n", + "if nargin<4,error('at least 4 input arguments required'), end\n", + "if any(diff(tspan)<=0),error('tspan not ascending order'), end\n", + "n = length(tspan);\n", + "ti = tspan(1);tf = tspan(n);\n", + "if n == 2\n", + " t = (ti:h:tf)'; n = length(t);\n", + " if t(n)h,hh = h;end\n", + " while(1)\n", + " if tt+hh>tend,hh = tend-tt;end\n", + " k1 = dydt(tt,y(i,:),varargin{:})';\n", + " ymid = y(i,:) + k1.*hh./2;\n", + " k2 = dydt(tt+hh/2,ymid,varargin{:})';\n", + " ymid = y(i,:) + k2*hh/2;\n", + " k3 = dydt(tt+hh/2,ymid,varargin{:})';\n", + " yend = y(i,:) + k3*hh;\n", + " k4 = dydt(tt+hh,yend,varargin{:})';\n", + " phi = (k1+2*(k2+k3)+k4)/6;\n", + " y(i+1,:) = y(i,:) + phi*hh;\n", + " tt = tt+hh;\n", + " i=i+1;\n", + " if tt>=tend,break,end\n", + " end\n", + " np = np+1; tp(np) = tt; yp(np,:) = y(i,:);\n", + " if tt>=tf,break,end\n", + "end\n", + "```" + ] + }, { "cell_type": "code", "execution_count": 43, @@ -992,13 +1060,13 @@ ] }, { - "cell_type": "code", - "execution_count": null, + "cell_type": "markdown", "metadata": { "collapsed": true }, - "outputs": [], - "source": [] + "source": [ + "# Thanks" + ] } ], "metadata": { diff --git a/HW6/README.md b/HW6/README.md index 1ec0fd2..5639560 100644 --- a/HW6/README.md +++ b/HW6/README.md @@ -41,6 +41,85 @@ b. with Euler's method c. with Heun's predictor-corrector approach -**5\.** *Coming Monday afternoon* +### Paper airplane Mini-design challenge +[phugoid +models in Python](http://nbviewer.jupyter.org/github/numerical-mooc/numerical-mooc/blob/master/lessons/01_phugoid/01_03_PhugoidFullModel.ipynb) + +![Image](http://nbviewer.jupyter.org/github/numerical-mooc/numerical-mooc/blob/master/lessons/01_phugoid/figures/glider_forces-lesson3.png) + +**5\.** + +Using a phugoid model, write a function and script to analyze the flight of a paper airplane. + + +![eq1](./equations/eq1.png) + +Use primes to +denote the time derivatives and divide through by the weight: + +![eq2](./equations/eq2.png) + + +Ratio of lift to weight is known from the trim + conditions ![eq3](./equations/eq3.png) and also from the definitions of lift and drag, + + ![eq4](./equations/eq4.png) + +we see that L/D=C_L/C_D. The system of equations can be re-written: + +![eq5](./equations/eq5.png) + +To visualize the flight trajectories predicted by this model, we integrate the spatial +coordinates. The position of the glider on a vertical plane will be designated by +coordinates (x, y) with respect to an inertial frame of reference, and are obtained +from: + +![eq6](./equations/eq6.png) + +Augmenting our original two differential equations by the two equations above, we have a +system of four first-order differential equations to solve. We will use a time-stepping +approach, like in the previous lesson. To do so, we do need *initial values* for every +unknown: + +![eq6](./equations/eq6.png) + +where we are now using a superscript $n$ to indicate the $n$-th value in the time +iterations. The first differential equation, for example, gives: + +![eq7](./equations/eq7.png) + +The full system of equations discretized with Euler's method is: + +![eq8](./equations/eq8.png) + +* Assume L/D of 5.2 (a value close to measurements in [Feng et al. 2009](./feng et al 2009-paper_airplane.pdf)) +* For the trim velocity, v_t=5.5 m/s. + +**a\.** Create the ordinary differential equation function that outputs: v', theta', x', and y' +as the function `dy=phugoid_ode(t,y)`, where `y=[v,theta,x,y]'` + +**b\.** Plot the position of the airplane over 20 seconds if the initial height is y(0)=2 m, +initial position is x(0)=0 m, initial velocity is v(0)=10 m/s, and initial angle, +theta(0)=0 rad. + + **i\.** Use an Euler approximation with timesteps of 0.1 s and 0.01 s. + + **ii\.** Use ode23 + + **iii\.** Include the three plots on a single graph in your README. + +**c\.** Group assignment for **Extra Credit**. Determine the optimal initial angle and initial speed to +launch the paper airplane to maximize distance traveled. You can solve the path of the +flight any way you like and you can work with up to three colleagues. + + +Submit your github repository solution to the following Google form: + +[https://goo.gl/forms/QpINqnnEhMceqCy02](https://goo.gl/forms/QpINqnnEhMceqCy02) + +Initial judging by Prof. Cooper will be based upon clarity of solution (e.g. documents, files, results in README) + +The top groups will be distributed to the class to be voted on + +**Good Luck!** -### Mini-design challenge diff --git a/HW6/equations/eq1.png b/HW6/equations/eq1.png new file mode 100644 index 0000000000000000000000000000000000000000..de43d621e4b55087bdcfd273835e0c7b6d7887e2 GIT binary patch literal 4754 zcmZ{mS2!F{w}waWorpfVjNU~A!{~j~VTc;NkKPS}5ED^D^xh3oq9u9@A%tjA5)!?S z?sv}f-<*qcvDaGr+uys_&3c}_6ZCb}$cPz<0RRA*hB^ca0ANu()V{b60q~Mjat{C? z;&N72)_1nE0{{XS($gLF-@)jI4o>9$#VS%Oz0y~H3*wGUqcrE6uC7r0`$1pDExrw# zKQUfHh0;Lri}4e*38b-skIxn?JWDmqvg3VJhg?Pm1gOcFfJDg5&;kOfCgI zAX>(Zy0e*=r1--viHLI3A3J_^kGV{0%^gYKu^323xo&s0zH+H^S2GbD(n4eq` zkl#n$ha8}dWTH>!D~8?5^n8qhO|pQg*)2;%teIn{v|MTfJdiv6!ye5$f>O1Y<-%$n zKBM?xH-xtA$vwHGn1}@KFUO-FJFY40s`vL3j<8={zk>nuO#om>cgR{g0I?5HygCZd zbFZvV3kAsTABy8<|B{E@Q#6kdZXrn;ia6f$}%)-*LCSi^wIRtbA32SgdDRHQy z0QOPv1~S@6js}8V0^@>55|KRLh;@;=UaWAexi9Ws^rT2$MM9p>{6Or3FLJnGvv|@e zAcHy)^c4cg8BYdNCPy*-Cc^_4>Pw{IIw-b(^RFNT#a0!_RuEm`$r4ybc_<;jaXOK{ zkGqSL|3au8{jUML25YC!EeofkxBJjshEX@xcQrIn`k`GP?<)BPaVQgOTq#(AdBKb?Haj-g~1Lxw_Z^d)F%N?)TPPm6E#LybT# z;#H50V`blLE1nhY7Y^!k%+O0P2&Vi1TNDKSRYXY0l2}FxsYw?tPR)Jm`%82hLWSo~dQob#1{_NG1lgQ}NJ3lY2cg2UN$}7@qK)I8-Z@HVetGR{Mi5Uo$kzee% zL^L0lGf+IvFQF}wfxd-yl$e*sm28&$DRD35EF~#jEO}L`Fe2i>Y`^YsK8hHXOII0j z84DhHIl`LS#q*3)Ewd?YV~lR}Lq>=I#hB|@$e3rE0>>|5h3bNe)e=;Zex;YOjq2M% zr|QS$W@Ui}-sTBq_O-OtMr8<%f82(tBQ`uD{8?SvZ_%CTCZ9N;566+moy1xLdfa-Y zI)5rb;PhI?QuDHfVi432>T5b?>}n8F^TyP%dZ>a;|G+T+yR|c})9%dQnJin4rob9z z6B1XWcQww%*;TOmpcf>?13wbyWGnnknSMx3y_u@}(OrzQleOKy%{;>ZkFv7q=5SWI zO{B6+eV8Z1R+Rwll)|mD%<{QPjQI;41+$3y@ccV^92z7p5+BJMXQWKuC)elE*OT0w ze4#q4iZM+!m91Msqh0?n*|BV>ZzvB^=s5B0^E}fD)|StCpG~Z7p;P{KW?bp3%ed-1 zqJ_63E3A*%-NUha;HL9Zjl}RZ;`R4O5wLCZK?^Ah-CEn)MEGoj%aY~*XuN7d)9Gt7 z`A2y-!DfRNqNd8$AdhMnCYwd4XCqVV{ocA>R*MYtM7s-nn{AJlbe6`>DETRAlXjbzVAnL;<@5RY z{ zg*+yV#bF%Xiw?M+JLrLVz^y_ey?`5$4ZXER)zDfWy+2CE+Ou9CE&VK#L%-I&Z%0xQ zydwX}>QZZ5Ay{>bkU2l>lP&-BCOG-(m@|trTg^)1Li&`&fz90|&0WCdCRE^G)?pjw zXoaV#tnB6GX6mCh?qY{4$Z$dj@1CCj$L*FMA4L}ub)yU!%B}9Guf~wy>&inW8YY|b zj1?@t>$(c>M#@Sd;$`h*V=cj!$SwH98N$N9^}Y(T(|9tvRV}P;#n-OhmKEWB1f3}T zIrb#eu@3Tr;Q;pQ~)t_7tS2Kez|Qw@YaQ`{9u}_nrX|^d`T|gby)RD%fG!h z;qUIwe`$xivWc+#ZL{+)->JJ?-v45Zq<`fWSGUyE6L2ekaJRQM+lQpMVU=ShVxW>w z3Y0nHTbs%m%1+tJcVkgWc_dF4VScB52fIBm?`b8{iKQRh9lFZDmWvRZV2e0YTVNxh zxfC829_3Bv$tg=Jo6viv`zx${^G|}QYL{Wd*bBF_rpiol)a zjmhR-W5D1+@XP0p`CWRU3JRtd;LX0Hl--oDd~A802}eO|dXT(p*#6!6l>{w1m$&NO;X^@y(9lsKSR-Mgq8It~>CW##`KAE@8#%4) zpzPBc*wq}FxPY@P;5rGM)NrWO$^&@yK3@5fu3 z(Bx6+@Ku7ixYP&}Ua2H;vqCGI$>ZIxBPCwP^y4V$jtxPr26wnvL*#F^mO>nFp0CXN zY+l)Tx_Ke*&?mSz_sjtCJECp3ksjjcmr@a&I8)o+5ppRs!#h49ucFDIhU^l|W~l4hD;k(q7V7PRie4 z>*o|To7_mVX4^a}IUn%2#wz>gJU}=nJr};?($rUH#dKoZ0c+hH{G<8zMyPGIM!F8N z0P2I7KB_hs50Iy2DAq-KktY5>oto3Xe!ID_(Y3v|TerQn<>qGd2=t)JBe;aNyqbs6}pfn@+{x_jY!`_rxtSDw} z1F;;EmAi1~)dBJWOcPf%Ic1F0XX0n&p14LAZfeNM_LS1ti&aH(qTfh+A+PId?GVEX z*yz&vI!z7L?~f!IXxu+gMaSj-k|vKy_iL$fC=fk2&(K)C?gby(sIivDacu5_y=4bJ`%Vg-sNd8t z_(;2V-2XEk70RTJhl^%Mi?)f<4rj2>o;-aY?&e*n+}cVlV5uj8%({@6ca_=ba_vuC z*Kw=%*R`KSkMt^*bv>2!ERdO9aWiyq_9+S?T+45JLRD? zFWbo`$N9=O_N$qy1Ts!>m@wNUQ+!_>H}*dB>qbBV4~oxJv$`$xt{anXf?8)}es4L+ zXsM3P!>^l~N}_kwE*XKo0y?%g&ViKZC%0>XC5gwh;#70NrbL|lGi98W{$)hJIYP#y zo-`&G^-p&4P>L;5I7TM;)CHgnYw8!82KAk-BrO8#lsQ|H+w_>+fx=TJwb=D{1p(v& z-?u&=`fkzT5j0WU1g-VQ)()3feC{l2Y5fUH28{+FnxmRED};*cUajO2hing9vT&+P z4XMH8)y;Q3j=_&E+kDmGXy(;yIo9rYlm%~mnA1M|lC+|y2c9`q@ma8`1) z!>WhN45#doF@hMy{ir3Jb^@PV;OlX@V&Rl&FvvcS#r|A;8k`lB5#7$g5;HEoLn7i> zfKVsRf*o~*h9{Fo`<5^rIVfUV9nt)fARq6eBoN*2 zE=d_0nF2hyYArn2+`JSt7a77q`$iFkw_vof>l)ae!XFKePV#Z2xT);jE0 z6t*WtD|X{@k)7!eaISgbM!OYC9C#Q}m&Vvril6mGyyY$`QCfeT7JI%k3|$HLK4r|E zedUAYx6)nHN)K|(sH339tzin6W8KD*tZ{IxQ$Av_D`5Qc^eXM~a0giOuAmtC;@6q5 z4FR5*ULqel1SnZCxUtKqrru~}ezcr{gEKM}r2{k(D!}e&uX)~V1wUTSKw@ib>WBHE zwHnO4@%#x;MF%bRz{Fq+Sg)(_zY5S;pu!Y|y18;J9$}{dT6fzA-$U~7)>QW<0p?ze0M zh%U!ZOu(w{NnA7r|n<&__=*@z=sJ0?9K&D!Bj=beqgUk1ft{)enHJ zLDMob`Bp)Oe3Z)z<)+$&pzSTPw|1C{lnlZj&$-uet@Dd)%$y0bCAWb~?bn>>)VT^F zhL<4Wx-qBK@0Rt0%qYu7v&cwkv4HuGldL2f4qZUH_W`tfdZs!gwKRz0cO?FXvo~E& z9rD-J0$C$gk3ymeYILIq#(`WH+6ucmceF z9)7qb^(9sT$yXZi#FNT1I@woRIkp`XfE-eC1Vnvy7? z9RZFP6}fXY&n^l|0a7@WHGiT=YzI~$EQ0P{NGJ!Pe2T&c$;KzQJsMty2Ca3y+LdT< zZ^jzdr2^2+;1Zq#{PXbKb0bvD5cW0*2U$BGhld0Z6A_gb6crT|lQt3+lNA$}m5>n> zk(3n?$sKL=_&jCF2(-f=KZf*+ri(#4FI-9m34G9M8$w2BH}_K zB4Wg%^cfuw2;YAZLoZK+vZtG;kDv%Z7~$?EjBp4Pb9VQU@i0;I5cc##2-*8P^DjCJ QJTL$a6`#p-BwA1&~lgiaC!`QAMZKe&AB*h)}Ci(&Hk-t)?Rb5*Tk5a=rhq@rw0Il$=7HGd+*!kCc+J(TaMkcQjiqC917-|EUrechXYs63i*%w!XrV0wTZZ519VNa-3ba zf3l$Ax~{gL$pQkxq4(GtYsEzK85-aGO%Ky*MQjq=4$u-jrl!i9i3APCoOyrjrGtGz zUEPAMw7Ir{?)gI60gBM-w0jDYYN!krSvfV;Gs$fGhppyeMnTRQq?WCvkxZ=A(W~F9 z-|Yz;^Z<{<5e@Ts*m>YawFNYsJX+M{QDEwAL9j{}NXmFWLC2TYLuMD&?-12JGuv#r zeMXb7|FBS2ALHG9S<3^DfbEDM0{ieN%xtV_J1^JMuEMCIw!AiaiY6M_vDkHl4!g0RvAc!zR!y zKwCx?#7e~$2DpSFtC-kB1*>TO(pctPQw$aDqcK`h1IM!4V}nup+vnOE4Py zFpM^_Pv{ZjtH`rR^;TL$_;M9x8O3V5M>N^xjc!MNVRZT;S94Fw>T)UZT^r#+d)4wNrH9BK z^&XoqtxDKwsL)&r4i42vLS*RDq9i+&e#^{7;*viGG(sJCv0?k4^kse3qiuc)ZM&&`>1`@59&+2?NSMtsdO=s#Ny#>e3^$kF7N2vPUvo?l!4;I}c}R}aC5T!%zXV}xm(*tHw)^~!9^bUY0HDqS+Y zQ-~8^6~C3@-Obg_+Z_jWKl@`*OvG}@kI9z-9MQ5(0Jwdn}n$-S3rGgt7Lox^A- z2WBQzbT}w9SUTu+UHfjn7dmLn*OqGSx@UO8+nOa+0#8%)lO%ObkfFIqyvV6YjYz48 ztN}e2tq!r(Sr~Ggvyh90Gdqtx4{H9xydlpvKQeD2Z$8g6Unrj;e>^WLU*jtT#p5!E z+Uv%5tETFGb?YH~efX6xsYz5tNI$J6<#!K9_nVi2QY=00J%K&g6b->?S&h=1qM5ws zxn{*UOGo&Nw~tCW3-1;9-|@1IDR3!gFSRJZ8!n67P5SC63Xx22LcFMMtgi8n^nSA) zy4^@`)L|-Onr}Q`4Ax35zn*VfFh&BK-!*@1-DBx~C$KER+O@Q^h~I4EZuZXyXzE9Q z2bTuZoeXRI%Xq99+%1yJ(4>qKo63Ou45W_X*b!Kfk2Ux3&HjY`is5Dw)oS|6=M|no zE@YU4W3wPy?=<#`ebU8yy8A)T50UtGu^_E*qU>(f3;Bcj_L}wRhgr)f9gSz?8aiJB~k6`Px7C*~j*Y5KB~1?g z+=NBK{$XFx@Uh}?8;?o!sWEtk>93UEL+&^Y=$ZpaVFNwD>NATMqXiVcJ4ur zIHo-o3}cr4@2ZFAuVPq}pNBuEakp`=8JR+gHOM$lM(n2SIm za{}3@fAY^k$J5ZK?{$$*kLK!x-OPi`vDHrAzS|FCO>S$&LWTr=LP`u~z>g5bf#Cw5 z{GVy+9cPT+<9NB8GlnyUGWe@cYIA=idJKqWF_|wYu)QaqrHX zUN?t4BX34w*MH^Ki-h!iwkj;|{BY%c2liO{hd`y`#^he;+T$Ro zllD|c-D~#!i*?Z^%LnTF!Mvz%)os;&$0HjpHW;KsU?@)DcW6~xd2XqBxwq-Owk2W+ z_s-tOE+1@nbi90`pcj=?EZ{cdi;pBR&B@qY-IjSNY_N~OfiNV>F>c+jU(hopj zb@$-)v$-P_Ue3#FOG762d84<)VCqHNB;agdZVUhfQ2+>i`oH~D=n??2Ak+0{ zJ^*l;WT>lU@n~x8)iie8g1z(Z^|5%2(IstKx>s&&ETQmN*6RYgJw+B2xxe;42|rC~ z&T+dg#KpZ6$$9z4ljb~P%!oxaKbww=wv#I3ANpYmQ|3(?7b?1&e7INl6V`;I9&h-V zkB`9Z2adL@gnrejH#cV=A3H3r(ni;IoyVP}8xT~$Ntg+NBnPADyfkmYW&Q}-#pRvw zrG*zffBX5a=xUUAI%?BhUC--Dg`QN<=h_{HGZKn$)|Ia%WAmmNDVMMH{&Sy}v^rk3 z7(?o;TBIX=fX{TsM<{u>;t%d6DZg}(Iioo(DfdZU-EK1i^XtbN49pUxPN>o|mgo*1 z;^AQoR5V9!c@**U!iFnISCkB}P2`5djt2S(5_ zRM&$UzJ?kmgsJK01L}gVWTx%dfdqwm3WFaNF?=qPvVeGsOieF*Ea*mueqMe>@(EWS z6ApMn^Q=1Ge@Ubv<;9G3G37HHuOV2TXt|QZJ-YBX@1>ouJl#>WkJ@)h^y^pM6+1Xl zRA?0DP0iZ4635x!gPeuDRp@ixn%znxn+cEpsr_NR-*am6_<#hQ>_6;jD zQxjT?yG()w!5guv30bJ=p0NpUsbtr%g`l$Wn<4QDQal(TJc)?OOD>Oq z5;}e?7ufCm`0+HAH|s3YYI@LF`}AkL9OOOumH9N;NPfx7ksL=qlmaY&%E?U$$Ludh<(j{Ue3iUu<#JgnSKqb)D?wPcFqs%vXna*V@4wOg-bB4RPM@PS z_DkxU?wHqatpl0K>>JX;Yns1gJX8CE$xuPDqnKiMg~zKauhWIDfDx?6*Awq>%&-O6 zq}+m~UR{+ZKblv3qf*piHTgI|rHVgM;?guXj^NJz?%uN<%=`hG%OCGg>M^}AKoG)>4#&J2nu|NHLK z=+&e?p6Uoz<;kV!8l?eBF0S4?YDtC1T`?#-j+ckbkNE z6uMQPoyj@6S~WSFeI32e{!_$JFH$?J(3IV#u&n5h+p?jm9tS@mG;;}SgbuZq8d$t2 zRMMS&{JwA6lxVCENAhXBgE^OXj*eG6G~~jiDjwIEJvjDZtX`Ni8J<-58p=@XOL01S z*@Mfw=Oj{t}2MOcYZc%aWl-!6!hq^Kv#^ZuQnF8{JI;rb&R@8j7PT#@Q0&o${rq5n1h0eO z_YJjFkaAsy_rHa6RdD{vb4{u05CHL7pXEW01Th>gj^yeAVXSrdQhBgBoN*#+u~2V7 zkI66ZwY(w*1efiYmDJ<3pBCd~rv_EV(bk4gEN%5qVP>%Z{wRpx_3q(a9Pucavp4EO z0C|Xw7jx~oIFfk0j3C*hA6=xkj(B6nEwNKVqMS=1O9(tEsbU10ucPy0rwbY zfHp@x*2el^T+kR-ZLB{aClB#gQkIlgl#^6~D&7BkyWk>Q?7!IvlrPEy&~m{#q2O2y zURq8D0u#_NHa3)#7l1$%WFQcEdO6U`h6_aezsOx27O#W#z~4two3MC2GsB zjkUaD8^}O$v2t3>CTbn#V$~MVDuSq}qq^c}{;2bwIafWbSC zG2c1Vw@^$ybOl?TVRZ*|ADP7@EpuP_9F%M8(qS5@ALf*m-@davo`aG&$^O0e)fRJ| z7T^*)p<=v9&D3KI$^^ zFlKVrNWeJ~L119R@)M}msm$Nrki!apr!~YG`tl&IjCXkZveV;4)To6&g80a`I+Q8Y ztzzj1`8afGOrN6p1Y#NBnvBVue;BT+zcZFgr9i4Reh)067LP7{3n`&FxeB4Ojr7sL zeHU<}&yG2dQR$$DMXeCX%C4*ty)wz0JKMIcm2MeEKbj2-R=jLS6q#i_q7CEZiHY6@ z1>d+r&Emk<7c14N-4IK~8V|iq=6uIAHdWw27c`Wxf1AX7TNW1vbugA=yH@zT8hSVX z%sA|!xW}E+o95Aw#8tI}yv=vr#{47favYLLW9l|jkh-bt-6FeV68&0DcgxFUV>hy9kwbl<)&pMFqzNd-vzUUL%b^Xz|5MzOo z;}L-o{D|Kz4YNW&_pqOV*5n(vyrUA|)hsJxxLRW$r>Lm0>zN8A2>lhR5yA^e>(X*i zYvMYb1i=q(7IQG&%q?ImP%?dC+FW2=7*nuVuu$M#C{RdOI9c$tP_-9~R8 zl+(0&J^C@dF1i+9nJq}pGk))D z?x6kRog4mUu~kv9iJ$eeBIj~8{KF!&-inY}O0R=37?jxxdr{p|UE?3)|9Tg@+d`|~ zWh7)&Xt3~JTs^J)R-tv#&wO!HGt)#@h19*D_Z>t#XZUAR_G? z+W6hI{>HJ`rd;~TtX?Y!mh^6^D5H4%J<0kX168Yt%E!6q?Bv&Q6gWzpNX$b`cA_%T zhuEG_n{cE(s!g&?v4m7iRabj1a5{1S*8Q#7&1B#vyeWLo00UEbm7PTkvoR?CI`T#1 ztH&4Z33Q!Eb7nY^%iG7bt?R7iSclH+Ir{nVjR^Rvb+?UzjbXWCc^qPt;4$@}OZ-de zzyr6hwTvHByd-N)>S$`-*N6DvJvbdE-R|`c&VTYV^tGGhn4nqzxv|)AW6EHv|1h@o zW%#?lSn|=1Z}0l%vhw4cYMq|vfA0O{o|VV5VH9@}X_v7UX2(9JwCPaLD9C z|Hk$=!Q~?R8S~5JsAMWnxM$7yFN8?5l*4rNLF$3S+P5cpn&cXPW;dy=Mlnt3elo- zR&~_%1pA~8ig0wBZxr+}4KqbmJNgIi+Q%6_P>%zT@&`ne>duP0!Ei%kMFEAw87f`p z^k3t-Ih?Y_vPQFbtIui%ob{bWkR;UiUj$F0bE4D0U_wd4S;GCszQ#%a9tA&zQSTUq zr>M;zc}+qQ{hus~%X{is>bg*;lHd3$<+i2|u^ZFJw8u7kJwZwvZeq+JX14hC+9~+y zwbkN@T*|=mDf7lPeFx}VIX^CMH!s@p@lJUzLqx?Z+eAolQz9neWLF?lAWO$i?nrT;8_DbKk?JksaTX@A zlDXYL+W941Q&i+~yqI#MK`0-20{#4~S!BZ~@WX1|*aw-PafXp*9L09$tS9}r;fmtW z0m8T1H|DA~!-k&H>sW{aIu_ytiMCa@#VsKQ4$wA%^%te2wW__oCA_q*ooJ(OLuQ2E zj_E+*kAATX*9z!A99!_2y+0naqo=KQt>oi~%ISv9;J+Tc#alxRc+Mr1^^|hZ^KkrL z-O5Jd$u@k1`hx z2a8I4u+o9(+~BL8tfZw}FK(@*8!Fc$tj}TR@V{Hu?e#PU(d^ypJtw)R$`P^yyb%XF zKY8h{9ZP?f?h{E9epM7-G+=bka3;KQap4m_)pGC7)rbdY@IC9jgCFf)7pv@7EhHswFpCHjP zettWuQhCWO{$e9(munNBldh=&0APdx0Q&?0PA*f}WdH~O1Hh_106;PUfZaEz$wc|` zgUnH12MRDjWL5GnUo^hD@JE;W?+TU<_-UVSUj|fYJp(POIXX@jcJNHodB7!xo9xY;uQi80Yl0GrsZ6aIFW^D!Oj>H4PJ8vzskFoD##6cm^2jLb*@#Fe|c(FPMArmIh3wtsvV%zkXLaKu+hbJm!!>CfI zsHGoHND?T6{^aop7vw>N{-P#W57X=VW!PFA3amdt##B$-MX-)GADU4U>N1bgc!YxF zR3zP&MIJ{RX@cln6Dw_-4X%Z+IgL(;N|@zO5~o4?3Ka|&kch1+{9BRIpbh72Zsw(U z`?O`7dsJSSLowmCxe$}FL@U&8o;QPqH9?kg8@sz15}3XKRib?@u^Icpleb{`_)2ex z#A$_Ke|2kBW;Aq^$ua8}dpvZvWfC_Yp+b-4yvq0~O%o?C#^LfeHo`WrJnpL9+69K?<^R9ORMzj+MxNi($w> zq!*y>jB-S3qkPbkGE!g&pQeFQ^Fx0 z03hW+LZP}yTU!9YcPb^tLHC_0ZTHTB^jd`61BEDE=v!XS(B%6t?jc;U-0CM?CFke{ z0=}4NF{S%2<=PE-s*Mz@D!I9BSD)G z&q@)=(4ElgR1;CY7l?E!A>kL&*L+##_U(FM8iBUyd2M^E;~!o5&oh=bewq6}A_Fk+ zoD);tgxv(}R2xWzA5Ikia4yvGFz`1@V@XP{pCM&R?LDO7Q27p0yw=@qQ@STZ3(u5rM-n1nZ?G#gS(dQJo(HV;E9U*SiJJS7XHXNrv2<@0@f zhlJXir6*djL%AWE_(7Z^D}kM;UUV}1kt*L%{2}YfeO3`{prW;|7|p|i*lNXRd6&9@ z2E0z9rB4haAaB;?j&iqix^&q_XvOFR62|1svi(-&Fk%pL^H4z*$=vC|@qx}&(!+pz zcRRmUg6*F;Xx;}=)==xmJ&Q(m!n^L-bP3m(@EhI5Ve2~ltvV~$CJ3B4j|q<-c#}L2 zy9#ArOmwwr8q@jGF}n}XR^p! zQJ0alzLM&YP$0@laf=75`)Y7JFpU1IC}hL$&Q}%g8{>yhwaYd>J( zFzJWUhwn8(s6>~a?aP>bw8z6U-ZQ#0>=P|boj9(?to2G|urjE#>ZUl6&dVYuIXaoH=0KpquCH$_TsY< z_7U8O`%?w|0(WHKls62x_0%Q&O-{|Kv_C^jba0Z49IcujXFTT>XALKgQ%IGJjs%Kr zx8(qD z{dDT}|7rh|DX9hYoLwcgCV8p%QO~FM0sK_GsNR5H*JN3?IU!kGcJX|EVy4p1z)^ajzljWkA^*V+UM!F|+QDe%7!RlF)H;WOXFXMy;HnU`K_TH$9&p7(t$W*k!NWlTRNG5BV%e5^GOxRJL0ZJl9+4jyJ<-O7ek zx{9GTPrBVF{m)81nh81o3R8>5OHN>xTCye~6~S58w7`d0LhM~ESEK=ywo|$j-Psmj z8-J=is(fOcWDF^vsjfz?(AzRDsV+ggsI(kG+o0!K{+co$GSbO3&9sVsjr1w}a_Uo_ zz|?a!r3G~|xS$v4W|m~O`CB(jx0JS6_|p6T^FKKo2pqc* zyJ$YCI}q3lS}a;iT^L`|*`wNX-OnPgq~fNye|Mc2P1elFK~FfJrRmqF$uC*Yd#v@*6YeqtZ&taiI{p4C z^i{ng+{SUSk;6$ZP|vm6#>0ExDn?sfJ_bC><`q(^I?wBs?^hfn zoo)J9`e-_H^<_=JorWD3;>2}ewi4B8_r|t=FupkcGX6zlPvf*lw}iXIs7s_ol1w0!zcRx46CU_5@MG;bd>gVn=vK{tdI0+g64oEtdoj+6;wAV4@+mkO*^mbL>BO{}IN_`}0ld$wt*d&p(`yss(qWYC~Fx`<`BZ z!DKH_szbS=CEbqc+`+Qb{OCoCMGJ5oUh$`4+xN>jLOMjGpE=}6Wr~^n z;hE46p&qUj(1*gf!hW6S+H*mTe^C2S@HWf90|^*aby(elcmg0`<0&k-15clSM6 z!Q8}V51x^f+iw$#t*V|D0N@V-0772@0Oz+yp{oFZ7Z?Cow*mklX#fDN+s7|2rEf0? zY&2990aOqXnY`PM)J@gY^H$&gPq-^z@7@-~7&R>=;stVgYFgpqmi{FG;4ZnEqP&6Q z>{f=WBRztlyTS}&r#q`cmXg`L21egUNy>ioyrKKx`6T6XAo_?)4Ux}KGxn#5&0T(sn(2QN>9~yHkL)qev6fo>xGJ&?;_ffTkihkpc z&CmXFly%SKye=w%R%9$jvix!sW1Z40VlXgB%ST>sIJM||_FYg;SN0hNO-rxlD^?{$ z?dyK<=W8LT99c>{eqez7nf>m;FoQ%jw53G_U9;40byoSN{)ZIz+vnlqaYi@%XC%T7 zwL5Yu8#zAn?B_77x??_*p2nlUcdstX_~S8w1-5CNjvuG7<$`$&Z}j;`Wzl$%;8O(PLK^fqk8rT-FT@Yd;ZZK z?lYEgbdGuF#m=We)v=E=?ItbRgUW_5ccLxh{g0ie{F2=QwW1?zj0sm< z*1_!+9OYNTh<@f6XgU4+hr07|wemwR5HIj!+$Hz0bHb0HM15I?(!57Ym0~av8Y%)?V>R6f%# zoUtqzH)xgpqAcYHjAb+NdUEx{YQR8JB_AxajL33Cr$1srFAc+Lt2X(_WtvJUEC%%S+CKB$Zb-Rvn|urWD_uNWz-XpnH?|A!XVSQZ%K*7OQ z24^xi&{Lqy3P1N^-WshwWvMWdaXBR`v!|**L)Te9(Cp+LGe}2@+iC;?*sfubSs7Ts zS1P=tf^zvz6M*o*eEsG9aOhMx1jt5O7E#sfItG#ZIv>?<6;Y{9ulRW=xQW0X4-}dB zn(%3G`|I4JjKC*kYm^@@ALD^T^2H27z1>=XwoxH?hGlxi0PpC;5I>CAE|5pOK{?x| zt?>uelUdFhL|JiQ8slVa9~zk4He(B|ab(oZK5i+hZ;7>2O=DcgX3Z1l5Z|3tw`6hA zKpz%l3udw7f%6VAdQ^NyOSwuR6C?_dcDn#8EN*D1+L#rEP{!qW$+>?~^6Y2B%e{&? z3}pLREPIZas-98$#4|B@2qS*cX|T<)B`b%^K&7CqR|b7z!*qM(VzZ5pMN+gkc7Y!a zyJ8Dn=48qv?W$}eh^v%>-zS8HNramxKeJXDMl!xiTb{o&YiSU$|4J$N>%lB(>jc?u z-kY)4Iow6R9rL_607n}ez9D*Q=1YD1KszR)waVrTupvy%h$E}kQ>()3zR))m5_YsCR6rP=S zmbH^(`DIjfU&$XNqkJOBN0qi4QKI84%C#A36#ZsUtOJth~kSivpjg9pr9HZqDeXew10QhX&r{v!iKSYVARDRP9Ve#-QNf<7Ha11jk}o zaOPlnlFkGlY)zNNZAx@bVlN)__YuDQst9>Q@Q=5FWWRiM{Pv~b#3-3y?5r^eh^+_W zRsbSkVMzgDVF3|I17Q(}h$uwtnE+TE0tQ1`ny3B`;NohBwDkPI0NMETx}4_ zu4s&aupk)10@cz|6Bc0sgGB|wU=cE5+V@Sj2={*weK%JO)YaM5LjVjA!nn8zVGzC| jNEZ(&w2=y0$khuYXy=XOn?~~AG5|FtZN+j0tMLBKJ{|KMd!3GC9kMqE9h=bMq_PsBlodw~vNAHlv9h9! zk}@jEJVti9?!CW%?jQH@etbUf`FySaw40v#plDGQpDjCAHfZnJMAZCe)|0~4gVSXl z?@g_P0#*=kN5hVzy-`ZCjH&(EdS0Slw>bshbLcC>Z)K%U$%xirF8UdXy?A&aibR%g z=Pvbp82U-1|8OCpKG#lJPRlc&RSBX6Ka(qT^6Ew>n#H>3m%iOznJK`^A3yoNHtH13 z#R%}r9MfI;m-sJctNyyioxM*Lg8}7Mq1U5txOSN0wT+bjTRXCrC)9 z$VGBrpl5d#9!gQ{F>Fbp<48q{Pop6M^Dmk*+8e&dX&F>XDkI zr>NNLvVO{yJT}R&^i|zIC7Z?lDA7rf3a>)Ow4P&8)QT=n1;*r*plGCC@?~OXoV>!Y z7sE_T?y3cQ@pzlFNnUDTw@%ee@g-yWncey!4R-Q3Xi50jJ<%@Y+TWk30wnfn_Bq1o zuO|LY5L?O)HqLCk z)^lO6qVG)7=uy+ZSez4_1~!c}6X&o=Sw=LCH@0c8Ak~!<54+8jBuO4?^@>iCu~PQh6^Ym)TSi3?)`^ zG$}Sonhd$3Z(SDR8#^2BK)rD#@J@Ph!wNi_za!;#mbNa)1TB#!@mHcjf+V42%*acR zz<0ZggH3seysSKhFF0RlpdX^!UO1EyUM#=(`694Ptc{|>>z%74ST3)_{9%22eM2ZA^x00rPCKJn zpOu7FnZ?hSvU)kySIQj9XG>+#*63Td!#4icVye<@y-5QVLZ~h4!jCS#G(PK-E0cL{ zCJm8Q{5P2VuRp5tEzPgAe-(9;skCqU{tT=l%$9FjZ6bZ5X1cSKdM$6Yca?vV7nA7h z+9~R5@b^BuQ`Y%=vTinrFwZFdRh~NXEL&L3Tz$R&Pd;~n@5bK|{j3%~lADrOEuzh}pFYWFGtGzrGxK!eM>)jdN($ZcAbB*Kc~ug@4s{pGlEfT-FuUT ziNlaEg>AlV40CV3x<-Hw;x-arLyd^ z1~ET(`d;hx@5w#3N11mr>HO{e8)m*?q}mi+zb605{^Pp#KB)vjt^X7O>VfyHlD)`L zi>iyF+PU!w|73gAO61*OA7txQLZL&ll`4L+v$D}Q_i(8e=20W&^6E{p`xXVaFoAQJ zcB9j}!@i`1q$VWB&1b1q+z%a##@4%qhVQuCw=~te51tYYi?1|Zko7Ugf0!;0EBlzM z-FL=3nkvBSoreJbcORQbt4KRd(`p@R{SrE$7NRy4 zNKm_r-TYkgMk0Q=_Xe?gpqag?4|}5UUig*D*4O=njjuQJkn`ZnfM!LOad z9MfV)Yo4^?+3D_H`Yh;IZBwC8`4^|LPa(sIZ_=aFj(5cJ#PW@tRSwnn1U!WT{jvk) z{Z3=$f9FwJ4z|BZHk6lp9W7^Zw@8$F9wP_uwMlJQg}+{HntlzLy>FRl%}aDX<2WA1 zf2<+KjMctxd~T!T_|ei|X*~g^hD(9D!;+o!obbOeV}Edt;m!Xl57z2-hklWijGd)h zja%~KL$=XlWuJ#-a=mMiH+i@0=XZbjElizsICoIbU|xM~*^K<_Cq&%(z(V5tg>{&N zM?8ol?Kb`1NWcB)Y=7STw!aWqV5P-9oVZqV^mVh_1`oO$K&Y8(cpLFod+Tgtaf*y* zJr#uWGw`x&r$%c0kzSm5I*_07t1v*oAcI?*E8gME{LKFEmc!d-28(1+|N6jj;R!q* zIwlnV$7oiF>GF}%pwf_3j^xww)bcT_tCsU|t;;`qN!hl$$`|8JPwjUdcK>{S8}M%x zbn(9AH~Me#XHKU#mN)7m`nRIJw7d&DtYUR^Y!CI8$=ez08F7VF+7@Hp3N9d7ZU4B< zv!!EC93&)UTSu`ZvprO1GUt445_UI6TL3_`Bmg8N0l@KjDPaWw!oUEq>H+|;JOBU% z7reO!KW|XEnHeDgRv1*f^gLq-HnzWY?jQfp2{AmQoCkC`6AJ^nMJ7IW5Tv4G>^lI^ zel!H&=5SuoCKJIlMnI-Qh+2taqgFt}{3n#2JqKrWij zx|;#7?)6rRV2;ENi!|NLRUlr&Op-34C~ED^hdfd3J5mX(^${K4Ha5O)-oCBe*4!M^ z=7`N`8bG{lY;I^!obS*mTndEb!<(VW0P#(aP3mzcOcn=1m_Tw&-kuET{ zP)Euy&x2pY1cUJaBietE!R$Lr&}VUa%D z$83%%5C95cD^)SFr}vgOP~4_P6i;2n8$d;j4Fa=!(V~`3;uQ|m20}EW54%MHNxrEgxvRN0@>}SEY5v_98*d#g!-h_f~nRZ^>7A_KzLQodjkA&gymer=&HaS;OYt5pE^~ zmLFQ(#ArL3rSjEZ;FBjfH5j!(a`Vx}mQF}67XGSc*0|?pBa?_d`L27ll$M#`oY(?= z&@y5YKoK>v_Ibn3L>BJ)w#w!a|1grON}_Lw8L+J+eIu+{70K)=`!bPLcX}!;Bkd|6 zb98tS_FS>4DQ&u?2M&JP`6w?!$Y$p*3eH%6LOm_-cQ*jUFRjc8`y?Ba$tw79jym8{ z{WagUc4MKBkh;nXj=60*VO8i)scMY&yPMmGrJci)!_3YvR{n6!OS$A~nmvxC)V)qM zqOJSvctjNyvP@(4t&rls!JJG#x3SZo@H{7F@Az6Zlk18#w{qt!FZx2*tA%1Xj~VaH z;OprU!52RFGT{daI=ckjSXSWd)yJUku21E522ptCtU`L}vaQFjDZ%}`9Pk=Ek8$;E z^&Hha;&GupEyIER&BdWE^|t;Zq_8SEJYjep#nItoE`$faA?tWO(R&0DWLB3Ay6VhI z>nrsI1l|n55@5Z$;qZ&h3U6l$i}55KyoDF<|JqruYxGj%H=g!BUu*-Yy*X@`GPB97 zPHu13sIKRCepUsU^ae%YSKASy%*mFmQ;r4UqDeQ2HRu&wYPcnm^OpdJYs(l}6dN*2 zLo}~nyuxM7HsUPfeB2o3qT8TI;U)^m7##BdRGlPB4JhPbsOkoVU2~tD%MB#OZ2mOC z{@G~#$ro+jGS&)7J7^ z8{H3+FN7Wos}XB6ZVMk<@*T^r>x!rnzIxHx5rbawCgwS9CMmB7xxEw|Gy?l~w2wt% zS`|Y@qL1i93z8T67iUKSN<=1?icWs($fe8zzVcUhO$k_<=qR?+-fYd$`A1yLM?N2l z$5oV59kEt0!Rg6ec?r02?N!PR3JLhjZIbVpPWDFjSV;!DCDrPeYmJgz83i`Ln#-;7r(^I7{7( zdav|g=9!34=Cz+u_{fUGju;EAgK*7@HjfitkfdAQ;{ zVeX-x=MI2^A?gYchyqmoIs^)XD#KJX6~L-6Fc{X+KKp-wK&*$aSJeLxm@UwFodcr( ztq^<*X6%dhyrqu~3-a&{^47;j0uZQLq_Ul@-BYC?f>)xa}N~{tsdujKv|a0oYIlFrb7B3|7K< mMnZi9Lp6eK7zHU|!*Gfo;l6TTeC5v>z{J23S)=cA=f40gvqc5~ literal 0 HcmV?d00001 diff --git a/HW6/equations/eq5.tex b/HW6/equations/eq5.tex new file mode 100644 index 0000000..96981fc --- /dev/null +++ b/HW6/equations/eq5.tex @@ -0,0 +1,4 @@ +\begin{align} + v' & = - g\, \sin\theta - \frac{C_D}{C_L} \frac{g}{v_t^2} v^2 \\ + \theta' & = - \frac{g}{v}\,\cos\theta + \frac{g}{v_t^2}\, v + \end{align} diff --git a/HW6/equations/eq6.png b/HW6/equations/eq6.png new file mode 100644 index 0000000000000000000000000000000000000000..47be3336944a5fea547748001c057d8f5d22fc34 GIT binary patch literal 5986 zcmb`KRZtwvlfV}V?m>eFcb3H$3-0b3oZu25i_2mmxC9UGEVgKH2p%jrK{gO1xJ$6( z`|AI2_i!&)_b}5lJ^dT$>aObfsHv`ii}eZ%007`BDavXC07xg#<95{N4&b*QL7NxBhY%&tH;)$FI!*7%4#JgG)$jgO=&<#l1g@wJMFu^gnyUUc74AsX@4Ysor6yX z2op9W>S$zPF2U}|J;(@^>Qy~~_T4zJQE6xhoh0~4;^c2Z++N&V)62@1o~Cc~{TSUU z!}x&|(vWV*$02T;iO<6;F80WgXJ*%{9jX#womtd(wJ0-l9e|nm*H(Kxf?FU$d1L*v_{C}kZyp8~G{+@o& zaH*(G4FG_TPWVtW*TrVfBGFLaAbHWD5Qnl)p?>Yb$UsuDz@`epJ^^&mVN|085TFo+ z0&GG}>u`xf=~ zL1;-g8qM7&-x`fh6f650*@jO$I+fv4k>f|)#p`Q=*C1$stfeMD@yn7=4YDFd_nHB^ z?2deu0Z=g+Hekruug=FZ^-P z`g`htcI5V|1k8AFLY-KVXor5&K^&_=UbrErK3WyDsn5@%zwXZ>vNO{a>J^a>hG^(R z2*XCIyS%W1DUcg8J&I#UV3T_z(mkcsw?o*R+AZ|DM*$&(P&vW9alNO$M-BQyCuKoc zjnNe@+$Sjgq9DyZ4yX)KVIM*{~xrx51Mpf(6?31C_Dl{xK+9>@-OrkQE57uNTznx{6 zt(x_CC9PB9;SjLwV~le6$|Wqm0KO~cN7We}lq~UvOi7z5j_H93&Q!(3qliU;Ap`BT zW&|pemr>x8=M@td3u`B7w-+0iL>6xsZxy?gFqB}I{49oR{QdfqoU##D?zdNom z?nZu2{@Nhf09>=u(BQO1X-&PUxG6J;ulA1lh*?z4PgNp2D-%oAM6GOm_Pg}B<9GQ* zuV$9^jKF>>7gzhPf%}eI1#F#9UY}-2gN^o#2TcS`)L~Yz7}L2r#}(xP_V1Mw%J0S- z@tP!@xf->aG2s<0zOGe{l$Jl=iH=NfeDhFuH~&eohn zGS@p)a@(}yAGcfmjG=K-4&~Ynvs&2a+V|(_X^(?=Zo29 zo!wK`M}m~ZutanxBPaO6nkh>=x8-WYW$LBn{$zNe42pEN3|XI8-(J66zwa-BU-S~` z!68rVeS(FYR^T7(4g8* zgolJT%bHrf+h}8S)DF;gYq0Y4IkSjSSC)za&e3}ZS1Kad->E`>%$ItX%%n>UJmQSU zQd3xG&S%bL(l*?~Cu~$~SZuG|rq=45`fcK^C#K`dx$*V*FnbhAG?Ogj5A;vI5?geFUqdQn z9n}BaNL&X}+kS88Yxr=taNKR=YHIEu;!d|2Qr8D7tkQ;gYHUgCsm{4KnR%PU1&r5Z zwLuBdqwtn!9ASFpT$K-A!dEB#;IgmresO=#7%~_#70mf>gf6IUX z+7l&9V{Ga6HL`Chj*ZsOe>)=P?mEpoQ5H;VS6h!D5012D$3JkZD0ke9SHwPi3amPB z{&N@~l=67Bt{UV-3#8s{we~67uen`4>eYjiML9>-tiZcJJV+cr9{!r^hvMJUh*4ou z5K6>CgfCftO=k~fChX)nQ_CfgN{|E_KdL?&JscZ%w_vJ8kPRLT{mr`*3kFTl245;H z(_+88M+lJS;zPHX}IIQdGl zQo4}k!(0<*Dg%n4^cAWhAE7WJNzDO5UD?J0_2bIKjpC0sVU=JmlP*igw{bZ85GIf zy%?Pf*V(>>LXd|qk;$xWXm8RE^*d695Zz@9>WKD>Q)LF9LERZ=M~tCAW@t3|2@)2( zcJSvtaD}NoI#Du8%{poXEq2V$4~oOASFHT7Z&YX-mK0OZP{o3=sjs;$d{S$4+K@zD zwr?kA+yKHnu{(kvc^dvPFs03ZYe}v+RVFwalFPNdq;A0jR8&ZbwMOZr`DY7Hiu*XI zqN{>3{cat^)(=+lC|a|@JNH$L5Au6#YG~6fHU-I4*rPW~ zOpIDIasZDrEi4K)pw3jdNDb$HD#&si@?WN|7i$l?MwCd*>}6`%{NexbIV8a(j=W^$ z#e%!gSv!!{1j+tRX_=5n`yH5NQkIvp#Tn)nH6M{zQf^%Q$sZISZeuLzvmP4t^CpQi ztb4@Z`>R2J1LZcH$1{goGYPxGd;suVs)dB~DKDXC(4E%@WD098{4KU=&<(=Q|02W&_uk-8R>Xq5W$GniBHFrL5tIn zFPr6x+{qF-W!Th`L?`pn#DSmd65c(D{)tDxER0RJzCe<)WNlrj{|N+>j7on&v6?#q z;l=Y|up{Kz0z|Yx%3T-u8EP$H(P!pZ%jGhxIc3+g(xNq+YrR8sn`Bl)B-LBS`IN;6mmXRRN){^z zJ1IhWbv;lAm*WlC1d*+GVRLc*eA~>$DkRvQBuSM(+oF{D+Ipd*Ue^7zzHd$079GN) zl>-Jm>Jf!CNpQ)E_YCOKyrDoOY=D8>2J0 zh>&mT!Kc?x;J2kaWzM0|w&3T9kTSgTw^?Qo2(B;r3fe-U7_02Q8Iw;TI~VRE7fb>l z`V&E0-m^pwI;0Qp3QCh>qp18>$q9BgBx%F>`ihsI8g67LLxa}RtFYTdNxye#!4Mh0 z#8v$oAL@j7Pw2ZLLWn{kJz$E~>M2Ge;s6(u^w@BuB6Q&XflpfZ;djva9$y0TL%c8f z_Mo0=t|y6DFtO;CC84A#pk>51F~xKBR7UXUdg6H$vDwl+M$OP}m{z)Aw!YJz9cjp6 zYLt|+tP$=eb89Jr#`&ec&cBd!&`T2dUXZ??Qyal#QqLK)SC)Qv-s*ScI_fjv?&%0? zy!a@1RjsRsu}C?? zm@B&(vL^*r#^CiKnc*fpQc3gSY|Y962~IrWZOi;To#ydFMHx~;o*pAuR~M1+p@gRy zXLYykjFt^X_?BX;!Ed$1@ockn2z%(_o-g@l(fXBNK7#;xMo(0#7>0Jj<~AH*q#u@R zO&rsPv9ztI8d!bj%&gnv%+?2!Z20}!Jjs?Vd}svI-2-cyomr);En~Jlm_hvwXyN0C zgPuPJ-^!Zz4-}q+ZQdiLrFo3@Cz21`hqRw=Mj1Ue<)M5C+3}ig?r)}+{QlKDnIs^2 zqVPf}wqDtc?~FN>HuEm$97u2y`%FB>y4)le*S|@&504_|a((eO+V)zB=xVx4 zqtoKI8jvJIzfo{U3`eEtc2(ynEHB~ySZc5JqbeSmtds4GL9oEtDFQ#=q~`R%LiRYX zPzN_q*3S3kM7sC!Mej$vXYjUG<3O)-guLfy0;eR3on+KIhKg2zZ4T;q1`L=eibXmI zTG3S63-L5sK4mYj*mnQk@1C)W@1Me(&zSh6y`}v=smzg2<#6R%_u2Q0--dWPM#xVb zJt_-$TR|5J$MwWpo8#9WjWViu`1h2q zu2e3)5i)a5b(Ov62Ol}ttrSAW=SW_qQRKHAm??fMP%?+=;ZB6KS<}_#?_tGlbcKJ? zVk{hXxqem6?Ea{0^2qt+t8`H-&o32v#)ATI^EJkOnb+vM1ip<--^*}$56QXQo<bFy@t@o6FKwmaezLl#indM7z4d$>J$&zCyix5{4_)M$P zpEwHzaoOFhVVxw`VWuQV`n%Sh^7k5HeiAoeLS!W8GZ3Mi6KSmYSp=86<768-vrImU zI3m?T2AQ||EMLW2*1H*6Cgq$fAVox(OGC%HhBGtQ5T{T?qNCPmj+E3cz%HiYJjgQDAZo_HGd4-LUVO= z1uUXPRs-*rEb2x;Vo<~Hlj)`bKZER8`JB;~o_19>hme>#M4R1gQ!{GwpkhslMtR8b za%NBmYEqy-nP8PSYHuhS?}}&Mh5|BbJY4x8d?I7f0d<X070;8-Wcr~ol^tb0-5^b=_ zI>)a!KUXqq-`J2za9FpkO1N00e4MyBAgKE^IlGN=9w?Y7!SMPg=|C|(Xq`qo@t{uQ z>8B0-iH%0Vq+T{YPVYn1T_a2_}>LdMDW%I z1OJjk>{4!!3Q)$m_1jj>g`=8odvaFYiEI&m?I)g2geSaP4|}}^0Tv*QsH@aazJ(b7 zDG&mlY=d0;16{s%D32Q*pJ8DQj2NLo=@6vNcsjl1nhT}f^lUR_ciJcmH-ZXUvGZ)zUpfmdZzZJM0{0`)JtP= zlXg7UWVfNY2Zr^qC)&jnf80YJ&$#FB3zL_ep_h%Nmo3=Z)Am^aKtNt0E?!Ivv{XJOGLEHAdw41l9jf1Pb zv>OBf;uD6506BO;LL378!eZT?Q_t2c|FKrJ^|5sZNZGhq*~+`QdU5e`1Hp7MYHCWn zAUYtBj~fUCVeyiEZhuBt|AXkbyLrjDIlFms0RcQ-F77;Dwh)kmi>I)wzJe={o3|IY WjgJGzPY2Ft2B0LTE?XmQ5%yo^3u15p literal 0 HcmV?d00001 diff --git a/HW6/equations/eq6.tex b/HW6/equations/eq6.tex new file mode 100644 index 0000000..d60386f --- /dev/null +++ b/HW6/equations/eq6.tex @@ -0,0 +1,9 @@ +$$ +v(0) = v_0 \quad \text{and} \quad \theta(0) = \theta_0\\ +x(0) = x_0 \quad \text{and} \quad y(0) = y_0 +$$ + +\begin{align} +x'(t) & = v \cos(\theta) \\ +y'(t) & = v \sin(\theta). +\end{align} diff --git a/HW6/equations/eq7.png b/HW6/equations/eq7.png new file mode 100644 index 0000000000000000000000000000000000000000..b0d7553c27c436a141d6aa883bb6d1335e537bf3 GIT binary patch literal 4140 zcmZ{mS5VVYv%vpKZ&E}8(gH{eEg;Q^^iD$W1e6jW1QJT5BVcGEMG=H3N-xr+iFAYr zC?L{7KxsrA(qh%+ zH^y3?arIZl;^Snsm`qgLEkvs=HLEH`M4Z$VzVK(4aau9oNCoY`84Q~F!FZb0aCo+; z;;wX*|`872Q zchcuOhI;-KQ4d{-s7|-OEp`t@WR#M=r*tlsW$)T<8fg&fOw4QBSsu^CiJ!b$So>xl ze1jI?5lis_}dP}EMNb}ryyG{R` zszArJNJiopXqt_ZqqpcPDA zMi$CM#u5p*L?SC0SR?o=sn)41ve{%Kga&9035MQ0$SdRRUfv*jg0L#J&?hnQ)vb0V zay9EX`av*YV54#E8Ru^X3bh<#*;I0rYQunkF*PK*BpX&tb3y^5vX8{7 z69xp_>0ia1$0)Z`!ym6yUM;(_M)J%cYwBp-wpF}o80|M38mMqtjwC#L?T9v%lP4y6 zTP%?63N^DMxF=4sL%TkXiX}ml@2bmf)3{WDzq(>WiTiwqOnfqgP)$c;S=Q?XPpdWW z8SvV)dY@v<;F`v^%Lh1{vZ@d45@ z&3*_yCFxTo)Ro;0&LniLhS@CPP8^zq?4oz-lCH59x1=m3eCP;%K&o7sxaui*Kz_jD zPklG?UxdJXs<*bH4iYL!lO8M9Ew>;!ALIR^KB!sIfg2Zj_(^A0Vb)-q;Rt)ieMbK1 zEm-XVa~XAOlwyaxIz>*JcOq0j(BKw}McksMl#{rRSk>de_&|c9_L**OO06E!lv8sC+^4Xzk~~6o_T$3|Bc_p`d&YV9CI8K*b#7}1YtJ5I;}F>G7V4NbvX(nlOk`C zHIX~1zC9d0+&u}34oHnmF&?2-n`Yay%%ixYGIB(Rafb%KNv?IS&3xf!)*A++*)U^) z;^Ps4kn>OX!7R2N)=Ksm}DiA23E11fUEl~LiMRB>zqYirT zJxXa>Up;z*zq)?qNof(fE1;8Jle*A*qvy@b5OKy{Om9dpE>(qpMoOhLyLdMLS+4Or zZwp85=Q-}B>_ygvfhIn-PYPYiSxe0e@p>zQW+`7Cg`i>?E%4{n&DAx&F}`nhBX*l< z4LXekjS3Kd-a*vT%5N6f7Eb0tOwCOFta>dlCLv`>R&J%;#k|H_W?91z(B$sxBg-Qh zPI@(gWn7kY81olp=saSH&HJGHba|a)@#C=KhgO_p@`Fi(6=SV=WNR6#pI5m?IFOMJ zj;;J?t$*>%_9>U|$?T(606aNosW820{N17LeT0g2`1?m$=OD7{1abl;K{&=-14L3H zVM%R?wTVaCquPg7DORwG>FR3CA5LfP1>FUWE=GjA(5BE`L@-?W^(!JR+zwInZKO~A zn@6Aa1intVDI<);<%M-??L2Ef)}b?dihnxH7H+d@+hr$jXISo39*-QY^qAJ~g!Gm4 z>$`uez4k%bQ=-3Z?%3Db*bkDpOtY%n$BzmURBl8!&4|EB(RT>BoCt3jszS_9Of)Uir(B?ZmN z&G9RzheupMI^=W3onS+dYFS^I9!QC-2V@cwe19LGfPe?}!NnEoAP0y{Kcv?*vRUV> z>Zmg+BC1Xk>Eu4&aLdCq)D&0k zzFIVqMd@EYZPBo1;HWuQ&QHkQ&5d??v{RnN5MFT<JhH<0%synI!PRF;}Y_Ld&kO*(^LPTXpd2Xp`xv$Y5bqn~Y_Xqojc8Q_i zDqb}Zn5kl~{owH^w8k2$(Pj&?|1c*8gvPTO0+$dfzk5n&lkF!wm zm5lBB!<}D3HHC$)$BQX!^@4e*6U{GAnuIru{6DPLjeU@wj5mxl<0x`CXF2I53|AC| z^jChbeQTj&H*AQJT91Iq1W=Bt399qc6k?()jZwCJJ;4Ru3VkqTY!r4wM1w0Ka z-K$&KNP6_*{9p$D2*V5IUaoicFIuZO{<+z1K>)>i##Bt#v<3WA-a6ly8zm7K&v=x$ zXgHXa69N_gip&kZ?j|NLWqERIC9^5t2)8|lpWFP~vTgfFgNO!ot#_YfohpUP^z(-Q z)tTg_yM8S7MXEVufnCFF9Q_65;`>|TO)+kD`TRSgzqIdrR#*4mM| zN(r`2_NZ1=BrN6U2D;OciDY%5`}*SptU{1>+x{M>|3<-Quu-&}|! zhHNz~{W@)ObKSfb5ISKxKjj&bI7_+wY9A;6LKrFQB|m&|Ent;KO+5Ma3MdMacuyc9 zoYfYz>GpS<^YMTXmsnRzkYTGW_v9K5h;_d4I*XU?PCb#&i~7&~7?26~*3FRRgNMg3 zGjWktopecJ6W6q+%Zd(Ynl10PIm5K(GwdEk3C-M%w|bql`)Hy$e-d)iQD)4d^bDT&Q8<4we5v|~(&vmTtnGc5a) zKI6u>lNK04${O=Bq-NvCpj3CPJ_GOk*U&_$J4QSE!=C8JrG$Q7mbdx8%}NKpd50%; z;j5?>ejtYO)*fj#&(Hrn>&Q$9mizIz(X{ENzge~dpEKi33~QDpEYq*F-u{V9n?WD* zx3Q)vMhi(5-(YG-PhFlX*x`leo*L*;$;B?F5dc0#yj%69r6zkqzRM_4QR^LB{*+X8 zbQ9;eJ*8a&dlkc=2SeXe6PHH)wUZUo8@IHU7L?;0`;T*+HD5UTy9(GFkvdv9rJCkF zIQ=$FnSvR*Dp^zXS1Y0&xtyKBn2^JiZTj!~ceG`RKNUp75?-QhK6tATpCPq}Vamni z9@-hS)*NaXtZnG%;OxVVDr&P#dI7ViI?VqBOow_hnSSNCSmtP{6s^sDb!dFVYt|Q1 z1ztJHvu9<_lWu%; zSg=c@Ya@g3qumuoIl448M>D?d)9^tzTr$SyIcnZTpn6pLkxG~&+|MB3@Vsy|ggfGx z>pSY?1l5l~$Oge&!ca)biIuJv%r6SKf%wNG2-v2c|9%mEw>xdNqv$>81YO->B!wrK4AIF$O$FO@BppPlcU@zxy~2m`kO% zw^&DhDLoRNqvBcoK^*haSFX@+A`v9N?Vp{ubnsdI$W}a@X4@ z>v#krfzD%Ba_G15v@)RD9*T#JBWNo8Pk6@6{Qfj|Q?#!0OAEl~%ie2*J!m?rPfpz4 z`0UYtPpsG9=Yu>sy4| z{y};}0u7Ba7aTqFuhbgtIn-lC^Lm^{{GMHDQtF+U8h1C3Dt(&`Uf81?85Av<^wn-p zq^_ijv#IY17$E!Q{odB3{aQks!$aF;vi0Q5LExOFIE-(J`o8F9n^Y~|9g}+L>aneQ zL^rqB^~uZ*MvM&(hK+<{C8onxB%-H2j)Bf+I_%Br*f(dW*Q&Rm5_{s8Kd~TQ%NpYRL>{(k^3oD14D=>G>;hSbC^ z0sj9~@b-i0qD@hL>bQqk7c|yQ9Tx~l%PR!RK*eOFWyIuVpu4{gE{6#q^xtea${*zk zsJY;rP}(>wUP4+D3Il5(5PFJWDD<`@6e>ge=`6Y7k`VbXVdjm)Yv4R_z7kMC3h(7D ng+~R-puKz*v6ebmDcnQ6q>DdVYzqCKt^qwQL(K~H2ao>;t?aPa literal 0 HcmV?d00001 diff --git a/HW6/equations/eq7.tex b/HW6/equations/eq7.tex new file mode 100644 index 0000000..2c15c8a --- /dev/null +++ b/HW6/equations/eq7.tex @@ -0,0 +1,2 @@ +$$\frac{v^{n+1} - v^n}{\Delta t} = - g\, \sin\theta^n - \frac{C_D}{C_L} \frac{g}{v_t^2} +(v^n)^2$$ diff --git a/HW6/equations/eq8.png b/HW6/equations/eq8.png new file mode 100644 index 0000000000000000000000000000000000000000..71da24fb59ff18e413320594e9ce84b036c9f4c3 GIT binary patch literal 7104 zcmZ{oWmMEr_wRqh07JKQOLv!mbji?NN)9ok0}Q2ugh;1=NJyt3-4fDW(j}lscME@> z|9#%vyViYi&faIA?`Q2dd+oE=`JkhvijPB$0{{TNx|)(60Dw9k{T%ei4G54SkOKg0 z7AHkT9Vc5`06;FJq`-8NOey*fPo@8c$&o0$*HKL1VnrnrnQ=^(70YdB>L|k_8qv5T zBSe&mbmck?IjfD7zEyE>*vO0bGv}1k7!yB9rZ~%-@LQcHyv=U9y!-teb|7>4nUDf_ zCt*U;TFcH>fZLk6mliDFsd*IAb>+lKtF0}56zBgOFJ}wsj&XI#tfW+UocgP4a&W5% zYZ8R2PBjtcma$JK0amPyT(1FK-Hm$U)9ZP z5K|tG(3CoTAP^N25J6vs9sk^QiDOcEco>D5uC{IZ19LS1V?cYrdJ2Fz0OW3teYIUn zDw6{Nse>b7^z>Ef>636wbU6@$5lj}$HIClgft3b&@d}p~g?j|FGh&s41Bt*S!GK+` zMHN07in$812Qkbc7eTR&<7kCwdC*&cW;SQj6Ze6`xJg3 zF58Kv`R;EOS~+O98=eMk>1sbRlc3fLd%YHj6n{+D&Avu(g%e0aA0BqZjU)$Q5nD43 zM(}m1G)6#3qLrA??1c3qlAoTdaZkpaF!^ZISZFk z8iq;5?#P|z9(?K5VV6sjcW+~#1>j+JcT@>D zP{K5c*a&Kf^`j*toVqP~@oahpYfN~IFv~*fy8K^tSN)wsgR@?sUy%4LVyoD^*wi==my`faF3)$zg7t=I``nq4%IdBwHHVDS8jK7cj#}j80B9Pmt(bu zN_2@SV0=mOh!N00zF;9SjQFj@Z^Prs{p}qx5*Z?)a;KJ?SgUTKhs7CEn}LW%q@P8c zebi)giFY07+(ewXd68aoT~l3$P(n4equI%s>XpmQ$`#>id!iWHx+)9JEl zPih7I0#B#F1s^l;KDFz+*e^BP(*Cq<5pNP@Q- zkpFdp@pHe|xU{<)0$EByQ_u6>K&_y?(Jk}Ui)>q7+bEuy^^~wp|ac_EiI?YDC23byP zgzI9ERPL0QUTRwD_Yya+=V#=sRI#@)%}=q3*om({+w;J?X*)l5Xs4+xf~~CEnVppX zMG{*kKGu_dgR-w?+?PLvsYSCTmu4@a&rRM|zRA9)0F#EGhhT=VhZ`tTbW3-;b$^ek zjk!{pQMoiuG?x0hR9)?|MPp01p|+veO9*vjJ79YT_1Bcm_>_*LX$~zKnI2XcaUNEg zL)5dkqy=@;y1K#Id+u7VRdMyB5K&X)Z%ud1dd6uxQ0tdH5^B3 z3F>6wytTUZ*fk{$er{#XG}eob&ju!b{qWTCuv(;=!`@ri|J_Kw1YH`sh-ga+`tlD2 zp6ML>GPs_e7in8-8MfcpkGCQQ`cuVw2ujo-A}S{s|Y#pi7^(@&!DF9_Z6vMe{M z93DP>AWDjV7Y}hUb*Y(Mv0!iEvtAB6Pd>Na9ShA>1S@1HQglgo?Q}bI`>l(vGs~vF zMLlqJiRJRxNKJB8i+BkQa+QiWb1iWBzdS=kLpA+|HF?DAxh|leUt72?S+uI&eY@%j zMTOQYS=c!KYGQHL3)FM3w(;^gc@?RpAs;C)!|eUGRBesRQ8Q%nXQ6k&RH|&xJ>Ez( z9hGhR&-9sehU&YTQM(s*?Dm)L<11Az-FC6IqZ2X3F?TUCO@mE~UVUPoVl%GcV(;A# z=5w1_-wypSDyryfAg=FmzvUfctQ0w1zCi6SU*lYxpZ57m>^pK2aTAe6@6|4u-jePV z&1GZyR4f@b?Y^*9TCZRZ$vw#pvw3q|k&XZM>y@9U@AN@1$yB0ZfIvXvh4h?D%_Xu# z$w}+)mF#5@o&9iiSGDi$?BRD)Hw&u(ln3Jms;a9Zw@k0XOM6SfP;=|60lU;e7?*e)L zrX4k29te|T6NJ zXw_J4mf>^rDJ>WNJ(QFfB0|bmD$G*eGUSiN=sCjNr{SUWa`)Tm;GZ&nH7kxLwZ^o! zp2vEl1@l9ksj#m~FR2bqS5G&c*Jf_ptlGe{7M0752grZU3`K{N_+>PI+|8DxeWQZP zPV4{f$G%CrzgX3LoeUUgm?7P zwAfU{ve8J1bB^_ijK1`^KiP0P$iTAOv;&_%H@=ep-F+Ol7Z;R`CJP;f@xG$ql646>xc_xy zj}Y|qJbuoX8{g{1Ii2!2HZj_&=|KU&pA7&|p#X667(#6WfVThu?7RX1sWbqf@c7)U zEB$zZX7fT-2_TdblFfTOVSA{VzJ9bH|95$++#fwU5QI8Z8M2N`LrftgSetDM07Uod zO7aHCh5gK+Xj*%^zJ{=i#+CDWK{+s*uL8b8098!AL)PjDKS0rlc4~>j=s2Z6qB#W{Mbu*Hj23S#EHQUv zkC8Hvk%*zTS^|gIw_k8gO?+Dx)Rv?HVB-&iq?T`v)z2kVQw|_=Lz1$sSHp^o4Z8Yr zz$!p^lQ`HdEyw5De#klU6LQ54dZrtynahb&(yec5p=@TMNGnQS4mUi z!;gfCqvNLZ?wYTXN2ohCh0-)}!Qfp8W>*Y|yOK8yyN2#CZYnyAzPf)7E$i2eVRu)t zKv0V5&+aP;dU2UKsA6Ga0nrFM9|KD!Q?S8;@zEF?{wr22?WATVb9~SX@owL(n9t6A z`b`AY!$7T(lvsP#Tb7@PBh#lH2^u8)Y{WL=>e! z*{hZRP-#8?U2>0~xHQ>q>wzq3T-ax_^>Wb&O;`$7wJ()D!+IQ=qzujU&1=2-oi zq*5rNdZSxui6OvHc}Dx1APtK3Wt9dLCUQnxi~Z?=tuK&ux^T9W$>6p6Mth6AQf46$ zNP;g+NxcRm;%Cjr(Q(S1cP4@aljp&}Sj^E8V>a}Iiz#+hFOOeWHH)eksu73nil>hn zR0QhPr700`aZGJArSOxwnS%`sN;(w3H0d)s+`rl1CZLU1EV#zwvixmaE;u%ur%C+; zKjl+GNLFFg3?sN7woTMrzZwXh)sshCO8U@EU9jLlJ&oXPCw1mtgD@)TtH_D$Ohz4v zHCKSi4nqn5ie8{&$*0+U8Bc$==-I&nnTcALO^Z1(GKJ}R6>-Y4uBdf;=;r|^R0=_> zp8%?ImJVuhD)x#ck}C2XGfguzX{;*!3{N_MRh>^E!AJDQQ#LNbf542$Sw8R!PO4f7 z{;F72dw08A99paDkg2KUryOqiaj;!-okBc-bLT;QvSxtBgCPM2aj;Afwqp$qX@Wep zh^y;Ke4{9`kenR zgkQmlG9RU{0dB^w2mXG8x(PiJ-tImbct5#GzW$(SA=&QBG#peb*P7PR!_cu~YvO9kIq8w_Rf zY`#k}wxj=5Mt%|nlQky~iQ|9Ab^)ICt0TVdwC_vb^x&BR-J)tIul6S=5kidytgeRK zC}lIhAX;inrUQ)T#X*j=* z3FlSzv{>a`Mb^=E17|SxV!oNMwjUt~{!&o! z^PYhQ=hsHP{;BH@0c?A$5YTGJwpA7RhLg^7s2WMY?v}5ja?=5|D)4V^=5%##s0%XM))Si72ABmK-q&G=t88Q~b`FAY`6j{E_O+90(EO%x= zW<1^Jj#Ab!-DiDr-CNm_sLa~lyuf`Z9Kilm{dO={9j8=WCH-4%s#1SUcEr~(bd2&d z1%;KOfHEU%Z$ELG@tHN(0%pEal$((sHW%jS>Z#2(j+BEs>!i)0sFx>MnE2ke1^e*a zkKr(yo@(UrZq1#a_X9Jba%`)CmVO6G$G;e6)7IuobNS~aB}%>0{uhej1@NM=z~ng69DZ+s1}>VjnW7joQP4s;=N{VZtGD@vD5&0r z{^*3KE3QP%;{2o8Ho!&gf5qtVnMHfm;sUB#p3&$apWAX33#}!o!uXii3ri(OYZl-< z=2K+-1imk7!1i;`3LuHGM-I6GczF*6u#kJ0rJKSh6{>||&Kwcpy-Kr5g_uLm46I_| z^0Pj#+An7mXrOWxhIht?u=X8lBGuZ&l`5|q^*whsc+zvpPt_}uz0wZk<=?Te*R1d3 z`>lqx8!@Ga8I19u=ZW~->q7|Zz`h`9?@u)^+FHNSa>ca@l87&+$EBv0H545#sz!eK zjwSf&+@ZhqG>#q#a=^soqoa&?hEZ;Xk)8~9ES;Tx$r7a{<>Wfqhu3AWz<0eNjnIaP zeWGxnP{T|~yx7xBupvXJLe%W#*!V)%b!mxupA&#i1<7=UZ1eTDB#jdLw@ZWxI_vnh zEL)onfX2;i(!k|=9VkLZ3~9jW2dVR!G-knEisQv~Flqc{n~K_SBGVPKWhw0=tqdPK zM)pjKkSj+o6vU3{K%zU~KhmV!j^{&f96cqSgAwmRNB$`63qm6EpVJ`vk!(C37i)@QRT!BfOolv1Y(Jo>LCzv0 z9^lTS)8Yxy$dt&1_ZD+^rHgDHoq>d<)%LOpsDX9+!lHIh<{Mh)I^88vq2UH*4m)X# z9<6UQY@-EGKwIc#>V%2}GG6sbGo0Fe$KM6r1PcV$7z9~E>=JQ?JeB67b&W#>4M3aB z(rQU!G?-EUb_Pi#rz>;cDv)H90)hBjBapl5)1QICpkR8M*A5Pp(!?)dgOv=$xEg(vS)k|7GSw$%;X<5up|C< zEc+>wM0YJImI|SS`~oX^pN;dLf(}N=8P4h;XNh(Rp)vE+p!y=1poo#?1Wh8^*^)xH zWP|uWEq`7?A=Q0ZNr*EQTzAcA8qjvgOkWr52LIuj)l8BOx5UH;2AYo0B6bWZbot(? zLB}SPIWl_1s=l+s1fufzOM|ZzLeu6_q(<~pcN`fqdwPu`zt@77a6KyZ*`Vwp%(CT+ z{1jg(*`kgm=rK14fC*8o?BU!bBTKWK4G%#M!Y||~_fB1Ue$OY>KBy#}=0VOJm82_& z=~k>J7R%7_3W7{yaVm(YC_BGxE-y?iX^;+dh8i-#t{N%32xNd4_}%3vhy`v0Xb* z*JiF9G55e5QZb0;@?`O>K>tO4fYqR9uCJU(ZOMDtI?b0f^-9g2Hi_c9%%7b}R)fNa zFGK>57e)U1r>ozC-T$h6dq7XA8i?(!Jjh^o-5?I1lgwwu>zTPY_ZH0xbjk|M`dqwk zqp3irl`X1D{5I*fOnt6Z3w0wC=IRh5{#}1gb~ma?4#jfHkw7^9{G0*G)ZRuVkP=A;Da2rbBzA})D zm1Yy|V2LQr$EEmTk?toF>j_CuKU2sZ*$i&!5LC?fmp;bNIIt3INd6B6I77Efgh+<1IL z|23RWC@~u20cy9YVgwXOLci`M5Bmv+vssyc5y{sRH!z(WixOX}oLC==U*kuE{xYo* zROQduK`BtTqo>}6u`CKD+<)up^pK*NqRjzz%I7soLOcy0gnDci>$KoWk*>2uIUW-o zatyd`Pf(`b;`^28J$fVPIwlk)3$7uVha9C8^e=H^O`3upf|;w6yGcrA4>Mk*zoKL z#X^2?$SjG-*)9)&FkQr?O_Y+m4PZLemP1tnG$)wIPhsD0mGl4FsVzSl0pb6IW67Pj zF~eIfIK?=#BGu|$OvwvLtbm42BwE%bu1=n7qj2C9?f9#aZ~(W}Yt9Z2U+BqF6dUWJ zsE7d^bcxep<4IF=OP%-946nt;@aKP&Vl?6IvxMFCEH$nCi>pq3Vm3KX~Jr0kRa;pM7TiPM(%gUk0j*%vk@g{@J5GKi#L{cvBU} z`XGyNs5c;6ST6P8+Coic-)&s#WN5OqupO3GfD!ebrq$UDqBQK9m2OwMGnCxupiG(p;BTkm_+9iDOtJoCUGwnC?5jPpkc4k)ijXzQv z+)F5FtR z`c_)&=80sfBClP!ja69!Fv;`7NU6d0rMII=^&$uZ}q`d7Bq>cAIFz~RR1&o zITJdMa97C$;`osF*S}e^$lSBtW$}>Cp{{zr-oCzMCm2C%tvY}1t^W>SClcLy{2XCL zD4QVctP%E7wqEv+1`rYu6z3Hb;0N_WsdhjFckwQ+cUJ`Cbs&4%5-UvQBA1CfbC!R+JP*>Jc`l|5i-Twd+8WQOM literal 0 HcmV?d00001 diff --git a/HW6/equations/eq8.tex b/HW6/equations/eq8.tex new file mode 100644 index 0000000..063b880 --- /dev/null +++ b/HW6/equations/eq8.tex @@ -0,0 +1,8 @@ +\begin{align} +v^{n+1} & = v^n + \Delta t \left(- g\, \sin\theta^n - \frac{C_D}{C_L} \frac{g}{v_t^2} +(v^n)^2 \right) \\ +\theta^{n+1} & = \theta^n + \Delta t \left(- \frac{g}{v^n}\,\cos\theta^n + +\frac{g}{v_t^2}\, v^n \right) \\ +x^{n+1} & = x^n + \Delta t \, v^n \cos\theta^n \\ +y^{n+1} & = y^n + \Delta t \, v^n \sin\theta^n. +\end{align}