Provide complete solutions to the following problems. The completed assignments must be prepared in strict accordance with the format and guidelines specified in the Homework Assignment Submission Guidelines. Not following the guidelines in a perfect manner will result in loss of LOTS OF CREDIT. The guidelines will be strictly enforced.
Problem 1: Modified from #3 Problem Set
The velocity of a body is given by the following equation
v(t) = te-t + 1/t
where t is given in seconds and v in m/s.
Find the time when the velocity of the body will be 0.35 m/s. Use bisection method and conduct five iterations. Use the initial bracketing guess of [1, 8]. Verify that the initial bracketing guess is valid. In each iteration, calculate the estimated root, absolute relative approximate error, number of significant digits correct, and the velocity of the body. Tabulate your answers from all five iterations. Show all steps in your calculation.
Problem 2: Problem Set
The velocity of a body is gien by v = 5e-t + 4 + sin(t), where v is given in m/s and t is in seconds. Derive the nonlinear equation that you will need to solve to find when the acceleration of the body would be 1.54 m/s2. Find the solution to the equation using bisection method.
Provide complete solutions to the following problems. The completed assignments must be prepared in strict accordance with the format and guidelines specified in the Homework Assignment Submission Guidelines. Not following the guidelines in a perfect manner will result in loss of LOTS OF CREDIT. The guidelines will be strictly enforced.
Problem 3: Modified from : Problem Set
You are working for DOWN THE TOILET COMPANY that makes floats for ABC commodes. The ball has a specific gravity of ρ and has a radius of 12 m. You are asked to find the depth x (in m) to which the ball will get submerged when floating in water.
1) Set up the equation needed to solve this problem in terms of x and R. When p = 0.6 and R = 0.055 m, verify that the equation you set up leads to the equation given in Example 1 in Ch 03.04.
2) When p = 0.6, discuss the possible range of the solution x in terms of R using the knowledge of the physics of the problem. Which interval would x be in, [0, R.1] or [R, 2R]? How about possible intervals [0 -1/2R],[1/2R,R], [R, 3/2R], [3/2R, 2R]? With the above discussion, how would you choose the initial bracket if you were to solve the problem using the bisection method? How would you choose the initial guess if you were to solve the problem using Newton-Raphson method?
3) Solve the equation by Newton-Raphson method and conduct three iterations. Use an initial guess of x = 0.055 m. In each iteration, calculate the estimated root, absolute relative approximate error, and the number of significant digits correct. Tabulate your answers from all three iterations. Show all steps in your calculation.
Problem 4: #1 in Ch 03.05 Problem Set
Find the estimate of the root of x2 - 4 = 0 by using secant method, if initial guesses of the roots are 3 and 5. Conduct three iterations. In each iteration, calculate the estimated root, the true error, the absolute relative true error, the approximate error, and the absolute relative approximate error. Tabulate your answers from all three iterations. Show all steps in your calculation.