Chris jumps off a bridge with a 15-m-long bungee cord (a heavy stretchable cord) tied around his ankle, Fig. 8–37. He falls 15 m before the bungee cord begins to stretch. Chris’s mass is 75 kg and we assume the cord obeys Hooke’s law, F = -kx with k = 55 N/m. If we neglect air resistance, estimate what distance d below the bridge Chris’s foot will be before coming to a stop. Ignore the mass of the cord (not realistic, however) and treat Chris as a particle.

Verified step by step guidance

1. Step 1: Break the motion into two parts: (a) the free fall of 15 m before the bungee cord starts to stretch, and (b) the motion where the bungee cord stretches and applies a restoring force according to Hooke's law. Analyze each part separately.
2. Step 2: Calculate the velocity of Chris just before the bungee cord starts to stretch using the kinematic equation for free fall: $v{=}^u$. Here, $u$ is the initial velocity (0 m/s), $g$ is the acceleration due to gravity (9.8 m/s²), and $t$ is the time of free fall. Alternatively, use the energy conservation principle to find the velocity.
3. Step 3: Once the bungee cord starts to stretch, the system involves both gravitational potential energy and elastic potential energy. Write the total energy conservation equation: $E=mgh+\frac{1}{2}$, where $h$ is the total distance fallen, $k$ is the spring constant, and $x$ is the extension of the cord.
4. Step 4: Solve for the total distance $d$ below the bridge by combining the 15 m free fall and the additional stretch of the cord. Use the quadratic equation derived from the energy conservation equation to find $x$, the stretch of the cord.
5. Step 5: Add the 15 m free fall distance to the calculated stretch of the cord to determine the total distance $d$ below the bridge where Chris comes to a stop. Ensure all units are consistent throughout the calculations.

Key Concepts

Hooke's Law

Hooke's Law states that the force exerted by a spring (or elastic material) is directly proportional to the amount it is stretched or compressed, represented mathematically as F = -kx. Here, F is the restoring force, k is the spring constant, and x is the displacement from the equilibrium position. This principle is crucial for understanding how the bungee cord behaves as Chris falls and begins to stretch.

Gravitational Force

The gravitational force acting on an object is the force due to gravity, calculated using F = mg, where m is the mass of the object and g is the acceleration due to gravity (approximately 9.81 m/s² on Earth). In this scenario, Chris's weight contributes to the downward force during his fall, which must be balanced by the upward force from the bungee cord when it stretches.

Energy Conservation

The principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. In this context, as Chris falls, gravitational potential energy is converted into elastic potential energy when the bungee cord stretches. Understanding this energy transformation is essential for calculating how far Chris will fall before coming to a stop.