Physics of Bungee Jumper for A Level Physics

 

The Energy Transition Phases

Assume the jumper of mass m starts from rest at height h. The bungee cord has a natural length L and a stiffness (spring constant) k.

Phase A: The Freefall (0 < y  < L)

Before the cord tightens, the jumper is in freefall.

Forces: Only weight (mg) acts on the jumper (ignoring air resistance).

Energy: Gravitational Potential Energy (GPE) is converted into Kinetic Energy (KE).

Equation: mgh = 1/2 mv²
Phase B: The Braking Phase (y > L)

Once the cord exceeds length L, it begins to stretch by an extension x.

Forces: Both weight (mg) and the upward tension (F = kx) act on the jumper.


Acceleration: As x increases, the upward tension eventually exceeds mg. The jumper continues to move downward, but begins to decelerate.


Energy: GPE continues to decrease, while KE is converted into Elastic Potential Energy (EPE).

2. Key Physics Milestones

Point in Jump	Velocity	Acceleration	Energy State
At y = L	Maximum (Freefall)	9.81 m/s²	Max KE, Zero EPE
Equilibrium (kx = mg)	Maximum Velocity	0	Transitioning KE to EPE
Lowest Point	0	Maximum Upward	Max EPE, Zero KE

Common Pitfall: 

Many students think maximum velocity occurs the moment the cord starts to stretch. It doesn't. Maximum velocity occurs at the equilibrium position, where the upward tension exactly equals the downward weight and when the acceleration becomes instantaneously zero.

After this point, the net force is upward, causing deceleration.

3. The Conservation of Energy Equation

At the absolute lowest point of the jump (where the extension is at its maximum), the total initial Gravitational Potential Energy has been converted into Elastic Potential Energy.

The Equation:

Mass x gravity x (cord length + max extension) = 1/2  x spring constant x-maximum 
Note: It is important to remember that the total height fallen is the natural length of the cord plus the extra distance it stretches.

4. Real-World Constraints

In an exam, you might be asked why the jumper does not bounce forever. There are two main reasons:

Work Done: Energy is lost to the surroundings as heat due to air resistance as the jumper moves through the air.

Hysteresis: Bungee cords are usually made of rubber, which exhibits elastic hysteresis. This means some energy is converted to thermal energy within the cord itself during the stretching and recoiling cycle.