Question

Jorge is going to bungee jump from a bridge that is \(55.0 \mathrm{m}\) over the river below. The bungee cord has an unstretched length of \(27.0 \mathrm{m} .\) To be safe, the bungee cord should stop Jorge's fall when he is at least $2.00 \mathrm{m}\( above the river. If Jorge has a mass of \)75.0 \mathrm{kg},$ what is the minimum spring constant of the bungee cord?

Short answer

Answer: The minimum spring constant of the bungee cord should be approximately \(118.6 \ N/m\).

Step-by-step solution

Analyze the problem and define the variables

The bungee cord will start stretching when Jorge falls 27.0 m, and it needs to stop his fall when he is 2m above the river, which means that the cord will stretch a total of 55.0 - 27.0 - 2.0 = 26.0 m. Define the following variables: - m = Jorge's mass = \(75.0 \ kg\) - g = acceleration due to gravity = \(9.81 \ m/s^2\) - \(x_0\) = unstretched length of the bungee cord = \(27.0 \ m\) - \(x_f\) = stretched length of the bungee cord = \(26.0 \ m\) - k = spring constant we want to find

Calculate the gravitational potential energy at the beginning and the end of the fall

Jorge's gravitational potential energy (GPE) at the beginning of his fall is: \(E_{g,i} = m \cdot g \cdot h_i\) His GPE when he stops falling is: \(E_{g,f} = m \cdot g \cdot h_f\) Where \(h_i\) is the initial height of the bridge above the river (55.0 m) and \(h_f\) is the final height at which Jorge stops falling (2.0 m). Then, we can calculate the difference in GPE: \(ΔE_{g} = E_{g,i} - E_{g,f} = m \cdot g \cdot (h_i - h_f)\)

Calculate the elastic potential energy stored in the bungee cord at the end of the fall

According to Hooke's Law and conservation of energy, the change in elastic potential energy (EPE) in the bungee cord should be equal to the change in GPE: \(ΔE_{e} = \frac{1}{2}k \cdot x_{f}^2 - \frac{1}{2}k \cdot x_{0}^2 = ΔE_{g}\) We can solve this equation for k: \(k = \frac{2 \cdot (m \cdot g \cdot (h_i - h_f))}{x_{f}^2 - x_{0}^2}\)

Calculate the minimum spring constant

Plug in the given values and solve for k: \(k = \frac{2 \cdot (75.0 \ kg \cdot 9.81 \ m/s^2 \cdot (55.0 \ m - 2.0 \ m))}{(26.0 \ m)^2 - (27.0 \ m)^2} \approx 118.6 \ N/m\) The minimum spring constant of the bungee cord should be approximately \(118.6 \ N/m\) to ensure Jorge's safety when he stops falling 2 meters above the river.