Photo credit: Matt Dinnery

The physics of bungee jumping is an interesting subject of analysis. The basic physics behind this activity is self-evident. The bungee jumper jumps off a tall structure such as a bridge or crane and then falls vertically downward until the elastic bungee cord slows his descent to a stop, before pulling him back up. The jumper then oscillates up and down until all the energy is dissipated.

However, what is particularly interesting in the following analysis of the physics of bungee jumping is that the jumper experiences a downward acceleration that exceeds free-fall acceleration due to gravity. This takes place in the initial part of the fall while the bungee cord is slack (i.e. not stretched). The physics taking place here will be examined next

The following schematic for this analysis shows a simplified representation of a bungee jumper and bungee cord, at the initial position (1), before he jumps. The jumper is represented as a point object of mass

Additional assumptions in this analysis are:

• The jumper

• Friction and air resistance can be neglected.

• The radius

• During the initial part of the fall, the extra stretching that occurs in the hanging part of the bungee cord as it supports more of its own weight, is negligible. This means that the change in elastic potential energy of the hanging part of the bungee cord (the left side of the cord in the two figures below) is small enough to be neglected in the conservation of energy equation. This is equation (1) in the analysis given below.

The following schematic for this analysis shows a representation of the bungee jumper and bungee cord after he jumps. This is denoted as position (2). The position of jumper and cord is set as a function of

Since friction and air resistance are neglected, the physics occurring between positions (1) and (2) can be analyzed using conservation of energy for the system, which consists of bungee jumper and bungee cord. This will allow us to determine the velocity

The equation for conservation of energy is given as follows:

Where:

Since the system is at rest at position (1) the kinetic energy is

The gravitational potential energy of the system at position (1) is given by the weight of the bungee cord (

The bungee jumper

Thus,

Note that this value is negative because the center of mass of the bungee cord is below the datum.

For convenience set the density of the bungee cord as

The kinetic energy of the system at position (2) is

The first term on the right is the kinetic energy of the straight section of bungee cord below the jumper (

The gravitational potential energy of the system at position (2) is given by

Once again, keep in mind that we are ignoring the relatively small mass of bungee cord at the bottom of the bend. Therefore this mass does not show up in the expressions for kinetic and potential energy given above.

Substitute equations (2)-(5) into equation (1) and solve for

This expression conveniently relates the velocity

The next step in this analysis is to apply the principle of impulse and momentum to solve for the acceleration of the bungee jumper (

To set up this analysis we shall isolate part of the system using a control volume type of analysis, which encloses the part of the system (consisting of mass

In this chosen control volume the bungee cord can be imagined as "flowing" across the system boundary (represented by the dashed line) at the lowest part of the bend, where the bungee cord tension is horizontal. The vertical component of force at this location is assumed to be very small, which is a good assumption for ropes and rope-like structures.

Since only the vertical forces acting on this system (enclosed by the control volume) will affect

The vertical force acting on the system enclosed in the control volume is gravity, which can easily be accounted for.

The next step in this analysis is to use Calculus to set up the governing equations.

Consider the figure below. The two stages, (1) and (2), show the "state" of the system at time

Once again, we are ignoring the mass of the rope segment of radius

Between (1) and (2), the change in linear momentum in the

We can express this mathematically using Calculus and the principle of impulse and momentum:

Where:

Σ

Expand the above expression. In the limit as

Now, the mass of the particles flowing out of the control volume must be a positive quantity. Therefore,

This means that

Substitute the above equation into equation (7) and we get

Now,

Substitute this into the previous equation and we get

Now,

The sum of the external forces in the

Substitute equations (9)-(11) into equation (8) and we get

Solve for

As you can see, the acceleration

Note that this equation is only valid between

To get an expression for

Once again,

For convenience set

The position

If we substitute

Even though this analysis is simplified, the result that

In the final analysis let's look at the maximum distance the bungee jumper falls.

Once the bungee jumper falls a distance

However, for illustrative purposes this energy loss will be ignored, and we shall apply conservation of energy to determine how far the bungee jumper falls, based on his initial position before jumping.

This analysis is set up using the schematic shown below. Once more we are ignoring friction, air resistance, and the relatively small mass of bungee cord at the bottom of the bend.

Where:

The maximum distance the bungee jumper falls corresponds to the lowest point in the fall, where the velocity of the system is zero. This means that the kinetic energy of the system at the lowest point is zero.

Thus, we can set up the conservation of energy equations for the system, similar to before, where position (1) corresponds to the initial (rest) position and position (2) corresponds to the lowest point in the fall (where kinetic energy is zero). The only extra consideration is that we must also account for the potential energy of the bungee cord as it stretches.

Now,

Note that the last term in the above equation represents the potential energy of the bungee cord, which is assumed to behave as a linear elastic spring.

Substitute equations (15)-(18) into equation (14). You can then solve for

For example, if

This concludes the bungee jumping physics analysis.

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