The Dangerous Sports Club, founded in 1977, periodically embarks on expeditions in which they participate in unusual, exciting, and frequently life-threatening activities. In this project, we will work through one of their unusual activities, bridge jumping. A bridge jumping participant attaches one end of a bungee cord to himself and the other end to the bridge. He then dives off the bridge, hoping he has correctly calculated the length of the cord and that it pulls him up before he hits the bottom of the canyon. One jump took place at the Royal Gorge bridge, a suspension bridge spanning the 1053-foot deep Royal Gorge in Colorado. Two jumpers used 120-foot cords, two others used 240 feet cords, and the last jumper used a 415-foot cord in hopes of touching the bottom of the canyon. Krazy Keith tried this later with a 595-foot cord.

Your job in this project is to find out which jumpers lived. How far did each one fall before the cord started to pull him back up? How far below the bridge did he come to rest? How hard did the cord yank on his leg? Did he survive or did he calculate the length or strength of the cord incorrectly? Newton's law "F=ma" says the acceleration of our jumper is proportional to the total force on him. Gravity produces a constant downward force. If the bungee cord is stretched past its natural relaxed length, it pulls up. Of course, this force is the lifesaver - unless it's too weak or too strong. A long fall before stretching the cord results in high speed, and air resists high-speed motion. This is the third force on the jumper. The three forces together with Newton's law tell us the jumper's fate.

25.1 Forces Acting on the Jumper before the Cord Is Stretched

Until the jumper is L feet below the bridge, where L is the length of the cord, the bungee cord exerts no force on the jumper. The only forces we need to consider before the jumper is L feet below the bridge are the force of gravity and air resistance. According to Newton's law, the force of gravity is

where

The force of gravity is equal to the jumper's weight, which for this project we will assume is 160 pounds. In other words, his mass is 5 slugs. This means the force of gravity is constantly 160 pounds (s. ft./.sec.²) acting in the downward direction.

The force of air resistance (F_r), on the other hand, is always changing with velocity. The force of air resistance is opposite to velocity (just as in the air resistance project), but proportional to the power, because the diver goes head down with his arms tucked in for maximum thrill.

where

The proportionality constant in this case is 0.1.

You should verify that at 100 ft/sec = 68 mph, there is a 63-pound force on the diver due to air resistance. How much is the force at 133 mph (not 133 ft/sec)?

VARIABLES

There are an awful lot of letters flapping in the breeze. Let's settle on some basic variables:

In terms of these variables, we have h[0]=H₀=1053, the height of the bridge. The velocity is the time derivative of height,

We write Newton's law

with g=32 and k=0.1. Before the bungee cord is taut we have the initial value problem

Use the BungeeHelp program on our website to solve the above initial value problem, just as we solved for s and i in the SIRsolver program in Chapter 2 of the text. We have solved for 5 seconds in the help program and found a speed of 121 ft/sec downward. You need to find the speeds when the three lengths of bungees tighten, h=1053-L, L=120, L=240, L=415. Adjust the final time of solution until you get these distances.

Figure 25.1: Five Seconds of Free Fall

If the cord becomes un-attached, how fast will the diver be going when he hits the bottom of the canyon? What velocity makes ? Give your answer in ft/sec and mph.

25.2 Forces Acting on the Jumper after He Falls L Feet After the jumper falls L feet below the bridge, the cord is being stretched past its natural position and a third force acts on the jumper, the force of the cord (F_c). The force acting to restore the cord to its natural position is proportional to the amount the cord is stretched past its natural position and acts in the direction that restores it to its natural position.

where

The spring constant for this model will be 3.4 for the 120 ft cord. This is equivalent to saying that for every one foot the cord is stretched past its natural position, it exerts a force of 3.4 pounds in the opposite direction. Longer cords are "stretchier", , for natural length L.

Express the distance that the bungee cord is stretched in terms of L, its natural length, and the basic variable h. Use this to give a formula for the force from the bungee in terms of h and the constant s

Figure 25.2: Forces Acting on Bridge Jumper

With no dynamics, how much would you have to stretch the bungee cord in order to produce a force of 160 pounds? At what height above the canyon will the various divers come to rest? Test your formula for F_c[h] by plugging in your answer to the second question and checking to be sure your formula gives 160.

The computer can give a piecewise function using the If[.] command. This is illustrated in the BungeeHelp program. Once you write a formula for the cord force strictly in terms of h (and the parameter s=3.4), F_c=F_c[h], this can be used in Newton's law to find a model of the diver on the cord.

% Use Newton's "F = m a" law with all the forces to write the equations for the jumper as an initial value problem of the form:

25.3 Modeling the Jump Now you are ready to use the differential equation solver in BungeeHelp to answer: