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--- gibson:teaching:fall-2013:math445:lab11 [2013/12/02 10:31]
gibson created
+++ gibson:teaching:fall-2013:math445:lab11 [2013/12/05 09:10] (current)
szeto
@@ Line 25: / Line 25: @@
 </code>
-Write a Matlab script to plot the trajectory of a cannonball subject to the above equations of motion and shot with an initial speed of 100 m/s at an angle $\theta$ above the horizontal. In the same plot, show the trajectory in the (x,y) plane of the cannonball with and without air resistance. Use your script to answer the following questions:
+**Problem 1:** Write a Matlab script to plot the trajectory of a cannonball subject to the above equations of motion and shot with an initial speed of 100 m/s at an angle $\theta=\pi/4$ above the horizontal. Produce a plot of the trajectory in the (x,y) plane of the cannonball with and without air resistance, both in the same plot.
-(a) How far does the cannonball go in x with and without air resistance, if you shoot it at $\theta = \pi/4$?
+**Problem 2:** Answer the following questions. You might need to modifying your script to answer the questions
-(b) What initial angle gives the farthest range with air resistance?
+(a) How far does the cannonball go in x with and without air resistance, if you shoot it at $\theta = \pi/4$? [Hint: set $\mu=0$ to compute the trajectory without air resistance.]
+(b) What initial angle $\theta$ gives the farthest range with air resistance?
 %%(c)%% If $\theta = \pi/4$, how much do you have to increase the initial velocity so the x distance is as large as it is without air resistance?
-Answering these questions with any precision will require some cleverness and a number of Matlab programming concepts you've learned through the semester.
+Hints: It's probably easiest to answer (a)-%%(c)%% if you turn your script into a function ''xdistance = projectile(v0, theta)'' that returns the distance the projectile travels as a function of the initial velocity and angle.
+In other words, create a new function  "projectile.m"
+<code>
+function xdistance = projectile(v0,theta)
+rho_air  = 1.28; % kg/m^3, density of air
+rho_iron = 7870; % kg/m^3, density of iron 7.87 gm/cm^3 == 0.00787 kg/(0.01m)^3 = 7870 kg/m^3
+C_D = 1;         % drag coefficient for sphere
+g = 9.81;        % m/s^2, acceleration due to gravity
+.
+.
+.
+f = @(t,x) [ ....];
+[t,x]= ode45(f,[0:0.1:200] ,[ 0,0,v0*cos(theta), v0*sin(theta) );
+xdistance = interp1(x(50:end,2), x(50:end,1), 0);  % This is how you can compute ''xdistance''
+% accurately from an ''x,y'' trajectory using interpolation.
+% It will return the value of $x$ for which $y=0$, i.e.
+% where the cannonball hits the ground.
+end
+</code>
+Once your function works, play around with the parameters ''v0, theta'' to answer the questions.
+On the command line, you can USE this function like this.
+<code>
+  xdistance = projectile(100,pi/2)
+  xdistance = projectile(100,pi/4)
+  xdistance = projectile(100,pi/8)
+</code>
+If your xdistance is a NaN, you may have to increase the number 50 in the interp1 function (1000 might be good, that is roughly half the length of [0:0.1:200]).

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