Problem 103 - Physics for Scientists and Engineers with Modern Physics

Question

Chris jumps off a bridge with a bungee cord (a heavy stretchable cord) tied around his ankle, Fig. \(47 .\) He falls for 15 before the bungee cord begins to stretch. Chris's mass is 75 \(\mathrm{kg}\) and we assume the cord obeys Hooke's law, \(F=-k x,\) with \(k=50 \mathrm{N} / \mathrm{m}\) . If we neglect air resistance, estimate how far 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.

Step-by-Step Solution

Answer

Chris's foot will be 36 meters below the bridge when he stops.

Step 1: Analyze the Problem

Firstly, we identify the two phases of Chris's fall. The initial 15m fall where only gravity acts on him, and the subsequent stretch of the bungee cord which follows Hooke's law. The goal is to find the total distance from the bridge to the point where Chris comes to a stop.

Step 2: Determine Energy at the Start of the Stretch

When the bungee cord starts stretching, Chris has fallen 15 meters. Calculate the gravitational potential energy at this point. The potential energy is given by the formula: \[ PE = mgh = 75 \times 9.8 \times 15 \] where \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the height.

Step 3: Hooke's Law and Energy Balance

As the bungee cord stretches, Chris's potential energy converts into the elastic potential energy of the bungee cord. Equation for elastic potential energy is:\[ PE_{elastic} = \frac{1}{2} k x^2 \] At maximum stretch, kinetic energy is zero, so we set gravitational potential energy equal to elastic potential energy:\[ mgh = \frac{1}{2} k x^2 \] Substitute in the values:

Step 4: Solve for the Stretch Distance

To find how far the bungee stretches (\( x \)), solve the equation:\[ 75 \times 9.8 \times 15 = \frac{1}{2} \times 50 \times x^2 \] Simplify to find \( x \). Calculate to obtain:\[ 11025 = 25 x^2 \]\[ x^2 = \frac{11025}{25} \]\[ x^2 = 441 \]\[ x = \sqrt{441} \]\[ x = 21 \, \text{m} \]

Step 5: Total Distance Calculation

Chris falls 15 meters before the bungee stretches, and it stretches 21 meters before stopping. Hence, the total fall distance from the bridge is:\[ 15 + 21 = 36 \, \text{m} \]

Key Concepts

Hooke's Law

When bungee jumping, the behavior of the bungee cord can be described using Hooke's Law. Hooke's Law states that the force exerted by a spring is proportional to its extension or compression, but in the opposite direction. The equation used is:

• \( F = -kx \)

where:

• \( F \) is the force exerted by the spring (bungee cord in this case),
• \( k \) is the spring constant, which measures the stiffness of the spring,
• \( x \) is the displacement from the equilibrium position.

In the exercise, the bungee cord has a spring constant of 50 N/m. This tells us how much force the cord exerts for each meter it is stretched. The negative sign indicates that the force exerted by the spring is in the opposite direction of displacement, pulling Chris back up.

Gravitational Potential Energy

Gravitational Potential Energy (GPE) is the energy an object possesses due to its position in a gravitational field. When Chris jumps off the bridge, he has GPE because of his height above the ground. The formula for calculating GPE is:

• \( PE = mgh \)

where:

• \( m \) is the mass of Chris (75 kg),
• \( g \) is the acceleration due to gravity (approximated as 9.8 m/s²),
• \( h \) is the height (15 meters before the stretch begins).

By using this equation, we can calculate the gravitational energy stored due to Chris's height before the bungee cord starts to stretch. This energy will later be transformed into stretch energy in the bungee cord.

Elastic Potential Energy

When a bungee cord stretches, it stores energy due to that stretch, called elastic potential energy. This energy is held within the stretched material of the cord and can be calculated using the formula:

• \( PE_{elastic} = \frac{1}{2} k x^2 \)

where:

• \( k \) is the spring constant (50 N/m),
• \( x \) is the distance the cord is stretched beyond its natural length (in this case, 21 meters).

The elastic potential energy gives us the amount of work done by the cord to bring Chris to a stop. As Chris falls and the cord stretches, gravitational potential energy is converted into this form of potential energy.

Energy Conservation

The principle of energy conservation is crucial in understanding bungee jumping dynamics. It states that energy within an isolated system remains constant. In the case of Chris's bungee jump, initially, all his energy is gravitational potential energy. As he falls and begins to stretch the bungee cord:

• The gravitational potential energy decreases,
• Elastic potential energy increases,
• Kinetic energy is briefly involved but becomes zero at the maximum stretch.

The equation used in solving the exercise is:

• \( mgh = \frac{1}{2} k x^2 \)

This equation sets gravitational potential at the start equal to the elastic potential energy at the maximum stretch. No energy is lost (assuming no air resistance and a massless cord), and this helps us determine how far Chris falls. By applying conservation of energy, we see how stored gravitational energy converts to the elastic potential energy, stopping Chris safely.