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?

On a particular day, a restaurant that is open for lunch and dinner had 127 customers. Each customer came in for one meal. An employee recorded at which meal each customer came in and whether the customer ordered dessert. The data are summarized in the table below. Suppose a customer from that day is chosen at random.Answer each part. Do not round intermediate computations, and round your answers to the nearest hundredth. (a) What is the probability that the customer ordered dessert? (b) What is the probability that the customer came for lunch or ordered dessert?

A box has 14 candies in it: 3 are caramel, 7 are butterscotch, and 4 are taffy. Keisha wants to select two candies to eat for dessert. The first candy will be selected at random, and then the second candy will be selected at random from the remaining candies. What is the probability that the two candies selected are caramel? Do not round your intermediate computations. Round your final answer to three decimal places.

When using strain gages during a mechanical test of a linearly elastic material, you measure strains in three mutually orthogonal directions (1, 2, and 3): ɛ1, Ɛ2, and ɛ3. You know the Young's modulus (E) of the material is 750 MPa, and the Poisson's ratio (v) of the material is 0.4. Hooke's law for a linear elastic material can be expressed in matrix form as: \left\{\begin{array}{c} \varepsilon_{1} \\ \varepsilon_{2} \\ \varepsilon_{3} \end{array}\right\}=\frac{1}{E}\left[\begin{array}{ccc} 1 & -v & -v \\ -v & 1 & -v \\ -v & -v & 1 \end{array}\right]\left\{\begin{array}{c} \sigma_{1} \\ \sigma_{2} \\ \sigma_{3} \end{array}\right\} \text { where } \sigma_{1}, \sigma_{2} \text { , and } \sigma_{3} \text { , are the stresses in directions } 1,2, \text { and } 3, \text { respectively. } Use Cholesky decomposition to find the upper triangular matrix U such thatа. U^{\boldsymbol{T}} U=\left[\begin{array}{ccc} \mathbf{1} & -\boldsymbol{v} & -\boldsymbol{V} \\ -\boldsymbol{v} & 1 & -\boldsymbol{V} \\ -\boldsymbol{V} & -\boldsymbol{v} & 1 \end{array}\right] b. Use U to solve for the stresses (ơ1, o2, and o3) in the material for three different experiments: \text { Experiment } 1: \vec{\varepsilon}=\left\{\begin{array}{l} 0.001 \\ 0.003 \\ 0.020 \end{array}\right\} \quad \text { Experiment } 2: \vec{\varepsilon}=\left\{\begin{array}{l} 0.004 \\ 0.008 \\ 0.006 \end{array}\right\} \quad \text { Experiment } 3: \vec{\varepsilon}=\left\{\begin{array}{l} 0.050 \\ 0.070 \\ 0.010 \end{array}\right\}

53-60: Conversions with Square and Cubic Units. 53. Find a conversion factor between square feet and square inches. Write it in three forms.

When adding the three numbers 2.12, 0.8 and 1.033 (all reported with maximal significant figures) how many significant figures will have the answer?

In Probs. 33-36 find the transform. In Probs. 37-45 find the inverse transform. Show the details of your work. \text { 34. } k e^{-a t} \cos \omega t

. The differential equation describing heat conduction, \frac{\partial u}{\partial t}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}, \quad 0 \leq x \leq 1, \quad t \geq 0 subject to the conditions \frac{\partial u}{\partial x}(0, t)=0, \quad u(1, t)=9 \cos (1) e^{-c^{2} t}, \quad t \geq 0 and u(x, 0)=9 \cos x, \quad 0 \leq x \leq 1 has the (exact) solution u(x, t)=9 e^{-c^{2} t} \cos x

Raina has a bag with 8 balls numbered 1 through 8. She is playing a game of chance. This game is this: Raina chooses one ball from the bag at random. She wins $1 if the number 1 is selected, $2 if the number 2 is selected, $3 If the number 3 is selected, $4 if the number 4 is selected, and $5 if the number 5 is selected. She loses $3.40 if 6, 7, or 8 is selected. (a) Find the expected value of playing the game. (b) What can Raina expect in the long run, after playing the game many times?(She replaces the ball in the bag each time.)

To solve the ordinary differential equation 3 \frac{d y}{d x}+5 y^{2}=\sin (x), y(0)=5 by Euler's method, you need to rewrite the equation as