Math Problem Statement
Use bisection method to determine the drag coefficient needed so that a 95-kg bungee jumper has a velocity of 46 m/s after 9 s of free fall. Start with initial guesses of xl = 0.2 and xu = 0.5 and iterate until the approximate relative error falls below 5%.
Note: The acceleration of gravity is 9.81 m/s2. (Round the final answer to four decimal places.)
Solution
The bungee jumper’s velocity as a function of time is governed by the following equation:
where:
- m/s (final velocity after 9s)
- m/s² (acceleration due to gravity)
- kg (mass of the jumper)
- is the drag coefficient (what we are solving for)
- s (time of fall)
The function to solve is:
We will use the bisection method to find the root of this function within the given range and iterate until the approximate relative error is below 5%.
Now, I will perform the calculations. math
Given constants
g = 9.81 # m/s^2 m = 95 # kg v_target = 46 # m/s t = 9 # s
Function definition for the bisection method
def velocity_function(c): return (g * m / c) * (1 - math.exp(-c * t / m)) - v_target
Bisection method implementation
def bisection_method(xl, xu, tol): ea = 100 # Initial approximate error xr_old = xl iteration = 0
while ea > tol:
iteration += 1
xr = (xl + xu) / 2 # Midpoint
f_xr = velocity_function(xr)
f_xl = velocity_function(xl)
# Check which side of the interval to replace
if f_xl * f_xr < 0:
xu = xr # Root is in lower half
else:
xl = xr # Root is in upper half
# Compute approximate relative error
if xr != 0:
ea = abs((xr - xr_old) / xr) * 100
xr_old = xr # Update previous root estimate
return round(xr, 4), iteration
Initial bracket values
xl = 0.2 xu = 0.5 tol = 5 # 5% relative error
Compute root using bisection method
c_value, iterations = bisection_method(xl, xu, tol) c_value, iterationsThe required drag coefficient is 0.4813 (rounded to four decimal places). The bisection method converged in 4 iterations, achieving an approximate relative error below 5%.
Do you want a step-by-step breakdown of the iterations? Let me know if you have any questions!
Here are some related questions you might find interesting:
- How does the bisection method ensure convergence?
- What other numerical methods could be used to solve this problem?
- How would the solution change if the jumper's mass were different?
- What happens to the velocity equation if air resistance is neglected?
- How can we modify the approach to account for varying drag force with velocity?
Tip: The bisection method is a robust root-finding technique but can be slow compared to other numerical methods like Newton-Raphson, which has faster convergence under suitable conditions.
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Math Problem Analysis
Mathematical Concepts
Numerical Methods
Bisection Method
Exponential Decay
Physics of Free Fall
Formulas
v = (gm/c) (1 - e^(-c/m * t))
f(c) = (g * m / c) (1 - e^(-c * t / m)) - v_target
Theorems
Bisection Method
Suitable Grade Level
Grades 11-12 (Advanced Physics and Mathematics)
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