Modeling A Bungee Jumper's Motion

Modeling a bungee jump is fairly simple given a basic background in kinematic motion and simple harmonic oscillation.  To begin, we must define a differential equation describing the net force acting upon our jumper throughout the span of their flight.  Assuming the jumper’s motion is entirely in the vertical  direction, we will consider two unique types of motion: free fall and simple harmonic oscillation.  While the jumper’s position resides above the hanging length L of the bungee cord, their motion is described as free fall.  As the jumper falls below the hanging length of the cord, their motion can be thought of as the bottom half of a mass on a spring’s oscillation, see Figure 1.

Figure 1:  This is a simple schematic illustrating the sections described by each type of motion.

In order to construct our differential force function, we must consider the net force upon the jumper during each portion of their flight.  While in free fall, the only forces acting on the jumper are gravity and air resistance, see Figure 2.  When the jumper’s position is lower than L, the jumper is acted upon by gravity, air resistance, and a spring force due to the overstretched bungee cord, see Figure 2.  Here, it is important to note that the retarding air drag force is constantly directed antiparallel to the jumper’s velocity.

Figure 2: These are the two free body diagrams illustrating the forces in each section of motion.  If the direction of the mass's velocity changes, the direction of the drag force will also switch.

We now construct our piecewise differential function

where m is the jumper’s mass, g is gravity, b is the damping coefficient due to air resistance, v is the jumper’s velocity,  and k is the sprint constant of the bungee cord.

In order to simulate a jumper’s motion, we must assign values to our model’s various parameters, see Table 1 for full listing.  As b and k are more difficult to assign, we solve for them instead.  When solving for b, we know that a person in free fall will eventually reach a terminal velocity v_t when the force of air resistance is equivalent to the force of gravity.  Thus, as

we know

An average person’s terminal velocity in free fall is approximately 53 m/s.[1]

Assuming that a bungee cord stretches about 4 times its resting length during a jump and that a jumper experiences approximately 3 G’s at the bottom of the jump [2], we formulate the relationship

Thus,

and 

Table 1: These are the parameters used for the real world simulation.

Finally, the differential F(z(t)) equation can be solved numerically and graphically displayed using Mathematica.  In the case of a real-world bungee jump, the jumper’s motion is qualitatively represented in Figure 3.

Figure 3:  This is a visual of the jumper's motion under real world parameters.  The jumper begins their flight at z=0.

Furthermore, we can test extreme cases to show that the model is qualitatively accurate.  Figure 4 illustrates the case of an extremely high spring constant k value.  Here it is evident that, while the simple harmonic oscillation section of the jumper’s motion is greatly altered, this does not affect the free fall portion of their motion other than propel them to greater heights as the spring force is increased.

Figure 4:  This is a visual of a jumper's motion when the cords spring constant is extremely high.  It is clear that the change in spring constant only alters the initial conditions as the jumper re-enters the free fall portion of motion.

This model could now be expanded to include more dimensions or could be used to prepare for jumps in new locations or under various conditions.

 Sources:

1. http://hypertextbook.com/facts/JianHuang.shtml
2.http://www.bungeezone.com/equip/cord.shtml