Numerical Methods

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?

An object following a quadratic trajectory h(t) = t^2 – 7t + 6 hits the ground at two times. Find the latest time it hits the ground a. t=6 b. t=16 c. t=0 d. t=1

6 The perimeter of a pool is 150 m. The rectangle at the right is a scale drawing of the pool. The length of each square on the grid represents 1 cm. Draw another scale drawing of the pool using the scale 25 m to 2 cm. Explain why your drawing is accurate.

Find the transform. Show the details of your work. Assume that a, b, w, 0 are constants.

72-75: Gas Mileage. Answer the following practical gas mileage questions. 73. Two friends take a 3000-mile cross-country trip together, but they drive their own cars. Car A has a 12-gallon gas tank and averages 40 miles per gallon, while car B has a 20-gallon gas tank and averages 30 miles per gallon. Assume both drivers pay an average of $2.55 per gallon of gas. a. What is the cost of one full tank of gas for car A? For car B? b. How many tanks of gas do cars A and B each use for the trip? c. About how much do the drivers of cars A and B each pay for gas for the trip?

5) Which one of the following points is in the feasible set of the system of inequalities? \left\{\begin{array}{l} x \geq 0, y \geq 0 \\ x+y \leq 12 \\ 2 x+5 y \leq 50 \end{array}\right. A) (-2, 0) B) (1, 10) C) (1, -2) D) (3, 2) E) none of these

The graph off over the interval [1, 9] is shown inthe figure. Using the data in the figure, find a midpointapproximation with 4 equal subdivisions forn \int_{i}^{9} f(x) d x

Find the transform. Show the details of your work. Assume that a, b, w, 0 are constants.

1. (20 pts.) (Problem 1.5): Compute the velocity of a free-falling parachutist using Euler's method for the case where m= 80kg and c = 10kg/s. Derive the mathematical model and perform the calculation from t = 0 to 20 s with a step size of 1 s. Use an initial condition that the parachutist has an upward velocity of 20 m/s at t = 0. At t = 10 s, assume that the chute is instantaneously deployed so that the drag coefficient jumps to 50 kg/s. (Show all steps)

Find the transform. Show the details of your work. Assume that a, b, w, 0 are constants. 1. 3t + 12