Question

Use Bisection method to determine the drage coefficient needed so that an 80-kg bungee jumper has a velocity of 36 m/s after 4 s of free fall. Note: The acceleration of gravity is 9.81 m/s^2. Start with initial guesses of xl = 0.1 and xu = 0.2 and iterate until the approximate relative error falls below 2%.

Answer 1

from the given data

by using the bisection method to determine the drage coefficient needed

80 kg bungee jumper has a velocity of 36 m/s after 4 s of free fall

acceleration of gravity 9.81 m/ s^2

initial guesses of x_1 = 0.1 and x_2 = 0.2

the approximate relative error falls below 2 %

function

by substituting the given values in the function we get

first iteration X_y = 0.1+0.2 /2

= 0.3/2 = 0.15

f(0.1)f(0.15)= 0.860291* -0.204516

= -0.175944

the upper guess and the root in first interval is defined as X_u = 0.15

second iteration X_Y = 0.1 +0.15 /2

= 0.25/ 2 = 0.125

by substituting the first and second iteration values we get

100%

100%

= 0.2*100 = 20%

f(0.1)f(0.125) = 0.860291 * 0.318407

= 0.273923

the lower guess and the root in second interval is redefined as X_u = 0.125

following below table consists of the remainder of the iterations

i	x_l	f(x_l)	x_u	f(x_u)	x_r	f(x_r)
1	0.1	0.86029	0.2	-1.19738	0.15	-0.20452
2	0.1	0.86029	0.15	-0.20452	0.125	0.31841	20.00%
3	0.125	0.31841	0.15	-0.20452	0.1375	0.05464	9.09%
4	0.1375	0.05464	0.15	-0.20452	0.14375	-0.07551	4.35%
5	0.1375	0.05464	0.14375	-0.07551	0.140625	-0.01058	2.22%
6	0.1375	0.05464	0.140625	-0.01058	0.1390625	0.02199	1.12%

after six iterations

a root estimate of 0.1390625 with approximate relative error of below 2 % is 1.12%