If you wrote the ball throwing script for question 2 in Credit Task 1, turn it i
ID: 3884601 • Letter: I
Question
If you wrote the ball throwing script for question 2 in Credit Task 1, turn it into a function.
Write a function file named DTask1_f.m and make the function declaration that takes v
and theta as inputs and returns the distance at which the ball hits the ground:
distance = DTask1_f(v, theta). The initial height of the ball is at 1.5 m.
We generally don’t want functions to plot things every time they run, so remove the figure
command and any plotting commands from the function. To be able to simulate a wide
range of v and theta, make the time go until 10 sec and add an if statement that will
display the warning ‘The ball does not hit the ground in 10 seconds.’ if that turns out to be
the case (use isempty). Also, if the ball doesn’t hit the ground in 10 seconds, you should
return NaN as the distance.
To test your function, write a script DTask1.m to throw the ball with the same velocity v=4
m/s but different angles theta = 0:60 and plot the distance as a function of angle
theta. The plot should look something like the figure below. You will need to run
DTask1_f within a loop in order to calculate the distances for various thetas.
Change velocity v=60 m/s, test if your program display the warnings and plot the figure.
Credit Task 1 -http://www.chegg.com/homework-help/questions-and-answers/steps-need-follow-also-add-meaningful-comments-code-write--start-new-file-matlab-editor-sa-q23238079
Explanation / Answer
% Simulation models of a TENNIS ball in VACUUM and AIR with and without
% spin using Symbolic MATH, MuPAD, ODEx solvers and SIMULINK
% This script executes all Simulink models and M-files except for MuPAD
% commands.
% NB: place all scripts and SIMULINK models in the same directory, and then
% execute this script RUN_all.m
% NB: Plot figures require from a user to indicate graphically where to
% display final computed data in the plot area.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clearvars
close all
%% MATLAB 2010 or earlier versions
%{
In VACUUM
x_z_vac=dsolve('D2x=0', 'D2z=-g','x(0)=0',...
'Dx(0)=v0*sin(eta*pi/180)','z(0)=h', ...
'Dz(0)=v0*cos(eta*pi/180)', 't');
x=x_z_vac.x %#ok
z=x_z_vac.z %#ok
%}
%% MATLAB 2012 or later versions
%{
In VACUUM
syms x(t) z(t)
syms g h eta v0
Dx=diff(x); Dz=diff(z);
D2x=diff(x,2); D2z=diff(z,2);
x_z_vac=dsolve(D2x==0, D2z==-g,...
x(0)==0,Dx(0)==v0*sin(eta*pi/180),...
z(0)==h,Dz(0)==v0*cos(eta*pi/180));
x(t)=x_z_vac.x %#ok
z(t)=x_z_vac.z %#ok
%}
%% Analytical solution of the ball trajectory in vacuum
% Case #1
%% In MuPAD
%{
#ICs:
V0:=25; h:=1;eta:=15;g:=9.8;
ODE_eqn1:=ode({x''(t)=0, x(0)=0, x'(0)=V0*float(cos(eta*PI/180))}, x(t));
ODE_eqn2:=ode({z''(t)=-g, z(0)=h, z'(0)=V0*float(sin(eta*PI/180))},
z(t));
Sols1:=ode::solve(ODE_eqn1);
Sols2:=ode::solve(ODE_eqn2);
Sol_CURVE:=plot::Curve2d([24.14814566*t, 1.0 - t*(4.9*t - 6.470476128)],
t=0..2, ViewingBox = [0..35.5, 0..3.35]);
plot(Sol_CURVE, #G);
%}
%% In Symbolic MATH
% syms x(t) z(t)
% syms g h eta v0
% Dx=diff(x); Dz=diff(z);
% D2x=diff(x,2); D2z=diff(z,2);
%
x_z_vac=dsolve('D2x=0', 'D2z=-g','x(0)=0',...
'Dx(0)=v0*cos(eta*pi/180)','z(0)=h', ...
'Dz(0)=v0*sin(eta*pi/180)', 't');
xt=x_z_vac.x %#ok
zt=x_z_vac.z %#ok
g=9.8; % gravitational accl
h=1; % ball hit 1 m above ground
eta=15; % ball hit under 15 degrees
v0=25; % initial velocity of ball
xt=vectorize(xt);
zt=vectorize(zt);
t=linspace(0,3.5, 200);
xt=eval(xt);
zt=eval(zt);
z_i=find(zt==min(abs(zt)));
touch_gr=xt(z_i);
t_touch = t(z_i);
figure
plot(xt, zt,'ko', 'markersize', 4,'MarkerFaceColor','y'), grid on
xlim([0, touch_gr]);
text0=('Ball is hit h=1[m] above ground & under eta=15^0');
gtext(text0, 'fontsize', 10);
text1=['Ball hits the ground in distance (of x) ' num2str(touch_gr) '[m]'];
gtext(text1);
text2=['Ball hits the ground after : ' num2str(t_touch) ' [sec]'];
gtext(text2);
title('Trajectory of TENNIS ball hit in VACUUM, v_0=25 [m/s] ')
xlabel('x, (horizontal) [m]'), ylabel('z, [m]')
%
%% Solved by ODEx
% Trajectory of a Tennis Ball
% Given: x" = -Cd*alfa*v*x'+eta*Cm*alfa*v*z'
% y" = -g-Cd*alfa*v*z'-eta*Cm*alfa*v*x'
%{
x(0)=0; z(0)=0; x'(0)=v0*cos(eta);
z'(0)=v0*sin(eta); eta=1; v0=25 [m/*sec^-1];
eta=15 [deg]; ro=1.29 [kg*m^-3]; g=9.81 [kg*m];
v=sqrt(x'^2+z'^2); d=.063 [m]; m=.05 [kg];
%}
clearvars
h =1; % [m];
v0=25; % [m*sec^-1];
theta=15; % [deg];
ro=1.29; % [kg*m^-3];
g=9.81; % [kg*m]
d=.063; % [m];
m=.05; % [kg];
w=20; % [m*sec^-1]; angular velocity of a spinning ball
eta=1; % describes direction(+-) of rotation;
% eta=1 is for topspin.
alfa=ro*pi*d^2/(8*m);
time=0:.01:1.5;
ICs=[0, 1, v0*cos(theta*pi/180), v0*sin(theta*pi/180)];
u = @(t,x)sqrt(x(3).^2+x(4).^2);
CD = @(t,x)(0.508+(1./(22.053+4.196*(u(t,x)./w).^(5/2))).^(2/5));
CM = @(t,x)(1/(2.022+.981*(u(t,x)./w)));
vacuum = @(t,x)([x(3); x(4); 0; -g]);
nospin = @(t,x)([x(3); x(4);
(-1)*CD(t,x)*alfa*u(t,x)*x(3);
(-1)*g-CD(t,x)*alfa*u(t,x)*x(4)]);
topspin= @(t,x)([x(3); x(4);
(-1)*CD(t,x)*alfa*u(t,x)*x(3)+eta*CM(t,x)*alfa*u(t,x)*x(4);
(-1)*g-CD(t,x)*alfa*u(t,x)*x(4)-eta*CM(t,x)*alfa*u(t,x)*x(3)]);
[tvac, XZvac]= ode23(vacuum, time, ICs, []);
[tns, XZns] = ode45(nospin, time, ICs, []);
[tts, XZts] = ode113(topspin, time, ICs, []);
% It is also important to find when the ball hits the ground
% (x-axis) and after how many seconds.
% For three cases: Vacuum, nospin and top-spin
% Case #1. Vacuum
z_i=find(abs(XZvac(:,2))<=min(abs(XZvac(:,2))));
t_gr=XZvac(z_i,1);
t_t = time(z_i);
% Case #2. No-spin
z1_i =find(abs(XZns(:,2))<=min(abs(XZns(:,2))));
t_gr1=XZns(z1_i,1);
t_t1 = time(z1_i);
% Case #3. Top-spin
z2_i =find(abs(XZts(:,2))<=min(abs(XZts(:,2))));
t_gr2=XZts(z2_i,1);
t_t2 = time(z2_i);
figure
plot(XZvac(:,1), XZvac(:,2), 'bo', 'linewidth', 1), grid
hold on
plot(XZns(:,1), XZns(:,2), 'm', 'linewidth', 2)
plot(XZts(:,1), XZts(:,2), 'ko-', 'linewidth', 1.0)
legend('In Vacuum','No-Spin in Air','Top-Spin in Air',0)
ylim([0, 3.35])
title 'Trajectory of a Tennis Ball hit under eta=15^0, h=1 m'
xlabel 'Distance, [m]', ylabel 'Height, [m]'
tt1=['VACUUM: Ball hits the ground (x-axis):' num2str(t_gr) '[m]'];
gtext(tt1);
tt1a=['VACUUM: Ball hits the ground after: ' num2str(t_t) '[sec]'];
gtext(tt1a);
tt2=['Nospin: Ball hits the ground: ' num2str(t_gr1) '[m]'];
gtext(tt2);
tt2a=['Nospin: Ball hits the ground: ' num2str(t_t1) ' [sec]'];
gtext(tt2a);
tt3=['Topspin: Ball hits the ground: ' num2str(t_gr2) '[m]'];
gtext(tt3);
tt3a=['Topspin: Ball hits the ground: ' num2str(t_t2) '[sec]'];
gtext(tt3a);
hold off
%
%% Simulink model
eta=1;
open('TENNIS_Ball.mdl')
set_param('TENNIS_Ball','Solver','ode3','StopTime','1.5');
[tout, SIMout]=sim('TENNIS_Ball.mdl');
% plot(tout, SIMout(:,1), 'gd-', tout, SIMout(:,3),'ko', tout, SIMout(:,5),'rh');
% title('Simulation of the Simulink model')
% xlabel('time, [sec]'), ylabel('Height, [m]')
% ylim([0, 3.35])
% legend('Vacuum','NO-spin','TOP-spin',0)
figure
plot(SIMout(:,2), SIMout(:,1),'gd-',SIMout(:,4),SIMout(:,3),'ko',SIMout(:,6), SIMout(:,5),'rh')
ylim([0, 3.35])
title('Simulation of the Simulink model')
xlabel('ground, [m]'), ylabel('Height, [m]')
legend('Vacuum','TOP-spin','NO-spin',0)
shg
%% Spin value influence on distance covered by the ball
%{
clear all
open('TENNIS_Ball.mdl')
set_param('TENNIS_Ball','Solver','ode4','StopTime','1.5');
ETA=0:.2:1;
% Plot label settings;
Labelit = {};
Colorit = 'brgckmygrckmbgrygr';
Lineit = '--:-:--:-:--:----:----:--';
Markit = 'oxd+s*h+^v<p>.xsh+od+*^v';
figure
for ii=1:length(ETA)
eta=ETA(ii);
[tout{ii},SIMout{ii}]=sim('TENNIS_Ball.mdl');
Stylo = [Colorit(ii) Lineit(ii) Markit(ii)];
plot(SIMout{:,ii}(:,4), SIMout{:,ii}(:,3), Stylo)
Labelit{ii} = ['eta = ' num2str(eta)];
legend(Labelit{:},3)
hold on
end
title(['Trajectory of a Ball with different spin values. ',...
'eta=1 (highest spin), eta=0 (no spin)'])
ylim([0, 3.0])
xlabel('ground, [m]'), ylabel('Height, [m]')
hold off
shg
%}
%% Alternative version and more compact version.
clear all
open('TENNIS_Ball.mdl')
set_param('TENNIS_Ball','Solver','ode4','StopTime','1.5');
ETA=0:.2:1;
% Plot label settings;
Labelit = {};
Colorit = 'brgckmygrckmbgrygr';
Lineit = '--:-:--:-:--:----:----:--';
Markit = 'oxd+s*h+^v<p>.xsh+od+*^v';
figure
for ii=1:length(ETA)
eta=ETA(ii);
[tout,SIMout]=sim('TENNIS_Ball.mdl');
Stylo = [Colorit(ii) Lineit(ii) Markit(ii)];
plot(SIMout(:,4), SIMout(:,3), Stylo)
Labelit{ii} = ['eta = ' num2str(eta)];
legend(Labelit{:},3)
hold on
end
title(['Trajectory of a Ball with different spin values. ',...
'eta=1 (highest spin), eta=0 (no spin)'])
ylim([0, 3.0])
xlabel('ground, [m]'), ylabel('Height, [m]')
hold off
shg