Newton’s Method for Solving Nonlinear Equations

Question #1Suppose a ball is dropped from a height of 2 meters onto a hard surface and the coefficient of restitution of the collision is .9. Write a script program to calculate the total distance the ball has traveled when it hits the surface for the n-th time.

SolutionStep 1. The coefficient of restitution for the case where the friction forces are negligible can be found as follows:

Cr=(h/H)1/2, where h is the bounce height and H is the drop height.

Since we know the initial height and the coefficient of restitution, we can obtain the bounce height, as follows:

h= Cr2*H.Therefore, the total distance the ball has traveled when it hits the surface for the n-th time could be expressed with an equation:

h(n)=h(n-1)+2*h(n-1)* Cr2

Step 2. In a new document in MatLab, enter a script as shown in the figure 1 below:

Function 1

Step 3. In the command window, enter a number of bounces (n): >>n=10, for instance, and after that enter the name of the script. The result would be the total distance after each bounce.

Question #2Modify the program from Question 1 to compute the total distance traveled by the ball while its bounces are at least 1 centimeter high. Use a while loop.

SolutionIn order to solve Question #2, use the while loop, as shown in figure 1 below:

Figure 1

The result, obtained after running the script, is shown below:

Question #3Perform the Bisection Method to solve a nonlinear equation until the error is bounded by a given tolerance. The given function is: f(x)=2×3+3x-1 with starting interval [0,1] and a tolerance of 10-8. How many steps does the program use to achieve this tolerance? How big is the final residual f(x)?

SolutionStep 1. A Bisection Method is a method which is used to solve non-linear equations where the initial bounds of argument are available (in or case this bound is [0,1] or a=0 and b=1). There are four input values in the program: f – inline function (f(x) in our case); a – left value of the initial bound; b – right bound of the initial area and tol – tolerance value (10-8 for this problem).

Step 2. A function body is shown in figure 1 below:

Figure 1

After running this function, the next result was obtained:

Figure 2

Discussion: from the figure 2 above, the total amount of steps is i=22, the answer is 0.3129, and the final residual is f(x)=0.000000005548982.

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