Final answer:
To calculate the spring constant of a bungee cord, one must use the conservation of energy principle to equate the potential energy at the start to the elastic potential energy at maximum extension and solve for the spring constant.
Step-by-step explanation:
To calculate the spring constant k of a bungee cord, we need to apply the conservation of energy principle. The potential energy at the highest point will be equal to the elastic potential energy at the lowest point when the spring is at its maximum stretch. Given that the bungee jumper has mass m = 55.0 kg and falls a total distance of h = 34.2 m, the potential energy (PE) at the highest point is PE = mgh, where g is the acceleration due to gravity (9.81 m/s2).
The bungee cord's unstretched length is l0 = 13.4 m, so the stretch x of the cord when the fall is at its maximum is x = h - l0 = 34.2 m - 13.4 m = 20.8 m. The elastic potential energy (EPE) stored in the bungee cord at maximum stretch is EPE = (1/2)kx2.
By setting PE equal to EPE, we get mgh = (1/2)kx2. We can solve for k to find the spring constant:
k = (2mgh)/(x2) = (2 × 55.0 kg × 9.81 m/s2 × 34.2 m) / (20.8 m)2
Calculating the above expression will provide the value of the spring constant k for the bungee cord, rounded to the nearest N/m.