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Chapter 6: Problem 86

The bungee jumper of Example 6.4 made a jump into the Gorge du Verdon in southern France from a platform \(182 \mathrm{m}\) above the bottom of the gorge. The jumper weighed \(780 \mathrm{N}\) and came within \(68 \mathrm{m}\) of the bottom of the gorge. The cord's unstretched length is \(30.0 \mathrm{m}\) (a) Assuming that the bungee cord follows Hooke's law when it stretches, find its spring constant. [Hint: The cord does not begin to stretch until the jumper has fallen \(30.0 \mathrm{m} .]\) (b) At what speed is the jumper falling when he reaches a height of \(92 \mathrm{m}\) above the bottom of the gorge?

Short Answer

Expert verified
Answer: The speed of the bungee jumper when he is 92 meters above the bottom of the gorge is approximately 26.62 m/s.

Step by step solution

01

Analyze the problem

First, we'll find the spring constant by analyzing the potential and elastic energies involved. After that, we'll use the conservation of energy principle to find the jumper's velocity at a specific height.
02

Find the maximum stretch of the cord

The cord's unstretched length is 30 m. The bungee jumper comes within 68 m of the bottom of the gorge. The total length of the bungee cord when stretched will be the initial height minus the distance he comes within the bottom: Maximum stretch = (182 - 68) - 30 = 84 m
03

Apply Hooke's Law

Hooke's Law for an elastic cord states that the force acting on the object is proportional to the displacement or stretch (x). F = kx, where k is the spring constant. During the maximum stretch, the only forces acting on the jumper are his weight and the elastic force, which are equal and opposite: Weight of the jumper = 780 N Elastic force = k * (84 m - 30 m) 780 N = k * 54 m
04

Calculate spring constant k

Now, we'll calculate the spring constant: k = 780 N / 54 m k ≈ 14.44 N/m The spring constant of the bungee cord is approximately 14.44 N/m.
05

Calculate jumper's height from the bottom of the gorge

Now, we'll find the height in terms of the cord's length when the jumper is 92 m above the bottom of the gorge: Height from the bottom = 182 m - 92 m = 90 m
06

Use conservation of energy

We will use the conservation of energy principle. Mechanical energy is conserved since only conservative forces (gravitational force and elastic force) are acting on the jumper: Initial potential energy = Final potential energy + Final elastic potential energy + Final kinetic energy mgh_initial = mgh_final + 0.5 * k * x^2 + 0.5 * m * v^2 where m is the mass of the jumper, g is the acceleration due to gravity (9.8 m/s^2), h_initial and h_final are the initial and final heights, and v is the velocity we want to find.
07

Calculate the jumper's mass

First, we need to find the mass of the jumper. Using the weight of the jumper (780 N), we can calculate the mass: m = weight / g m = 780 N / 9.8 m/s^2 m ≈ 79.59 kg
08

Apply the conservation of energy equation

Now, we'll plug the values into the conservation of energy equation: (79.59 kg)(9.8 m/s^2)(182 m) = (79.59 kg)(9.8 m/s^2)(92 m) + 0.5 * (14.44 N/m) * (90 m - 30 m)^2 + 0.5 * (79.59 kg) * v^2 After simplification, we get: 0.5 * (79.59 kg) * v^2 ≈ 28191 J
09

Calculate the jumper's velocity

Finally, we'll solve for the velocity of the jumper: v^2 = (28191 J) / (0.5 * 79.59 kg) v ≈ sqrt(708.52) v ≈ 26.62 m/s The jumper is falling at a speed of approximately 26.62 m/s when he reaches a height of 92 m above the bottom of the gorge.

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