2. A football has a mass of m,-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/m³. 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,

where p and up are the z and y co-ordinates of the centroid of the ball (z is positive to

right and y is positive upwards), up and u,, are the horizontal and vertical velocities of the

ball, respectively, W mag is the weight of the ball and g 9.81 m/s² is the gravitational

acceleration. The drag force is given as

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 p = 0 and y₂ = 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, p, 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 y₂ = 3 inch (half of the maximum diameter of the ball). Plot ₂ versus

t, yp versus t, u, versus t, and up versus t over the duration of the air travel of the ball.

i. What is the horizontal distance traveled? If the ball is thrown form the 40 yard line

of the offense, do the offense have a chance of a touchdown?

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?