Question

1. A football has a mass of mp = 0.413 kg. It can be approximated as an ellipsoid 6 in diameter and 12 in long. A 'Hail Mary' pass is thrown upward at a 45° angle with an initial velocity of 55 mph. Air density at sea level is p= 1.22 kg/m². Neglect any spin on the ball and any lift force. Assume turbulent flow and a constant air-drag coefficient of Cp = 0.13. The equation of motion of the ball is given as, dyp mp dip (1) Up dt (2) Up dt dup -Fcos(0) (3) dt du, mp - Fsin(0) - W; dt where Ip and yn are the r and y co-ordinates of the centroid of the ball (1 is positive to right and y is positive upwards), up and up are the horizontal and vertical velocities of the ball, respectively, W = meg is the weight of the ball and g = 9.81 m/s is the gravitational acceleration. The drag force is given as F = Column + e5)?D?; O= (5) Up Up tan- The diameter D = 6 inch. Assuming that the quarterback is about 6 ft tall, The initial location of the centroid of the ball can be taken as Ip = 0 and yp = 6 ft. (a) Using a Forward Euler numerical scheme with general step size At, obtain the finite difference approximation to the system of equations. Clearly write down the initial conditions for all variables. (b) Write a program to solve the system of equations using the above equations that will provide solutions to Ip, Up, Up and Up as a function of t. Make sure you input different parameters in consistent units. Use SI units. (c) Solve the system of equations with appropriate step size) until the ball hits the ground again; i.e when yp = 3 inch (half of the maximum diameter of the ball). Plot Iversu t, yp versus t, u,, versus t, and v, versus t over the duration of the air travel of the ball. i. What is the horizontal distance traveled? If the ball is thrown from the 40 yard line of the offense, do the offense have a chance of a touchdown (for a touchdown from this position, the ball would have to travel at least 60 yard or more)? ii. What is the maximum vertical distance traveled? iii. How long does it take for the ball to hit the ground? iv. How do you know all your plots and answers are correct? rsus

Answer 1

MATLAB CODE for forward Euler Method

eulerMethodProjectile.m

% MATLAB CODE for simulating projectile motion
clc; clear all; close;
% intial values/ intial condition
t0=0;
x0=0;
y0=1.8288;
u0=17.3858;
v0=17.3858;
%time step
h=0.0001;
t_last=3.50387;
N=(t_last-t0)/h;

t(1)=t0;
x(1)=x0;
y(1)=y0;
u(1)=u0;
v(1)=v0;

% constant values
m=0.413;
g=9.81;
Cd=0.13;
rho=1.22;
D=0.1524;

for i=1:N
% x,y,u,v calculated using euler method
x(i+1)=x(i)+h*u(i);
  
y(i+1)=y(i)+h*v(i);
% height calculation using euler method
F(i)=(Cd*rho/2)*(pi*D^2/4)*(u(i)^2+v(i)^2);
W=m*g;
theta(i)=atand(v(i)/u(i));
  
u(i+1)=u(i)-h*F(i)*cosd(theta(i))/m;
  
v(i+1)=v(i)-h*(F(i)*sind(theta(i))+W)/m;
% time increment
t(i+1)=t(i)+h;
end

% Plotting
% height vs time
figure(1)
plot(t,x,'LineWidth',1.5);
xlabel('time (sec)');
ylabel('x (m)');
title('Horizontal distance vs time')

figure(2)
plot(t,y,'LineWidth',1.5);
xlabel('time (sec)');
ylabel('y (m)');
title(' Vertical Height vs time')

figure(3)
plot(t,u,'LineWidth',1.5);
xlabel('time (sec)');
ylabel('u (m/s)');
title('X velocity vs time')
figure(4)
plot(t,v,'-b','LineWidth',1.5);

xlabel('time (sec)');
ylabel('v (m/s)');
title('Y velocity vs time')

% Answer for part c
Hori_distance=x(end);
fprintf('Horizontal distance treavelled = %f m',Hori_distance);

MaxVertdist=max(y);
fprintf('\nVertical max height = %f m',MaxVertdist);

timeTaken=t(end);
fprintf('\nTime required to hit ground = %f s',timeTaken);

Summary

Horizontal distance treavelled	54.434652 m
Vertical max height	15.996795 m
Time required to hit ground	3.5038 s