Project Help Notebook by Keith Stroyan

Help NoteBook Copyright 1998 by Academic Press, Inc. - All rights reserved.
Modify this Notebook to prepare your project report.

Project 25:  Bungee Jumping

by A. Student and B. Pupil

with technical help from Da LabMonitor and Help Notebook

Special Functions

Needs["Graphics`Colors`"];

My Project Introduction

Write this last.  Begin with the basic question and state your main finding clearly, but with few technical details.  You might even modify the title of this section to make it more interesting.  (The Dead, the Living, and the Jerked?)

A Sample Computation: Diving without a Cord

The student should read the first section of the printed project and complete the details of this section.

Newton's Law: F = m A

The mathematics of Bungee Jumping is based on the physical law that says the sum of all forces on an object is proportional to the acceleration.  The constant of proportionality is called mass.  In order to use this law we need to specify variables with units and parameters.

Variables

t = time since the diver stepped off the bridge (sec)

h = the diver's height above the canyon floor (ft)

For convenience, we introduce a letter for velocity,

v = (dh/dt)= the diver's velocity (oriented up, since h is measured up) (ft/sec)

Acceleration is the time rate of change of velocity, (dv/dt), or the second derivative of  h.

Parameters

m = the mass of the diver (slugs)

g = the gravitaitonal constant = 32 ft/(sec^2)

k = the air resistance constant = 0.10 (units??)

NOTE: Slug is the English system unit of mass, pounds measure force.

The "m A" part of Newton's Law may now be written:

m (dv/dt)

Free Falling in Vacuum

The weight of the diver is the force due to gravity, which acts down or the negative of height  h.

(Subscript[F, g]) = -m g

If a diver falls in vacuum with no bungee core, this is the only force acting and Newton's F = m A Law says

-m g = m (dv/dt)

(dv/dt)= -g  (Galileo's Law of Gravity for our variables)

We can solve this exactly as follows.

\(Clear[t, h, v, g]; 

    solnV = DSolve[{v[0] == 0, \(v'\)[t] == \(-g\)}, v[t], t]; 

    v[t_] = v[t] /. solnV[\([1]\)]\)

\(Clear[h, t, H, Subscript]; 

    solnH = DSolve[{h[0] == H\_0, \(h'\)[t] == v[t]}, h[t], t]; 

    h[t_] = h[t] /. solnH[\([1]\)]\)

\(H\_0 = 1053; 
g\  = \ 32; 
Print["h[t]=", Expand[h[t]]]; \)

Solve[h[t] == 0, t]

N[v[((9 (Sqrt[13]))/4)] 3600/5280]

You are going 177 mph downward when you hit the canyon floor.  But, do you suffocate from the vacuum first?  This is not a realistic model of a long fall.  We include the air in the next section.

Free Falling in Air

The force due to air resistance is

(Subscript[F, r]) = -k (v^(7/5))

where the minus sign is because ??????

Understanding Air Resistance

Use the air resistance formula to illustrate the amount of force on a diver at various speeds.  See HINT 25.1 in the printed project.  What speed produces an air force of  160 lbs?  What would happen to the speed of a diver traveling this fast?

Using Air with Newton's Law

Newton's Law "mA = F" with the sum of the two forces, gravity and air friction is

m (dv/dt)= - m g -k (v^(7/5))

We simplify, add the initial height and start form rest, as well as the meaning of velocity to get Newton's Law for our diver falling in air, but still no bungee cord:

h[0] = (Subscript[H, 0])
v[0] = 0

(dh/dt) = v

(dv/dt) =  - g  - (k/m) (v^(7/5))

We will work with an approximate (numerical) solution as follows.  Enter these computations and explain the questions below.

Clear[t, v, h, m, k, p, F, Subscript];
g = 32;
m  =  160/32;
(Subscript[F, r])[v_] := -k Sign[v] Abs[v]^p;
p = (7/5);
k = 0.10;
tFinal = 10.0;
\(\(soln = 
        NDSolve[
		{h[0] == 1053, 
			v[0] == 0, 
			\(h'\)[t] == 
              v[t], 
			\(v'\)[t] == \(-g\) + F\_r[v[t]]\/m}, {h[t], 
            v[t]}, {t, 0, tFinal}];\)\)
h[t_] = h[t] /. soln[[1, 1]];
v[t_] = v[t] /. soln[[1, 2]];
Plot[h[t], {t, 0, 10}, AxesLabel -> {"t", "h"}, 
        PlotStyle -> Blue, PlotRange -> {0, 1055}, 
        AxesOrigin -> {0, 0}];
h[10]

Where is the diver at time  t=10?

\(Plot[{v[t], \(-32\)\ t}, {t, 0, tFinal}, 
      AxesLabel -> {"t", "v"}, PlotStyle -> {Orange, Green}]; 

    v[10]\)

How fast is the diver going at time  t=10?

How does v[t] compare with the velocity in vacuum?  Why is the graph of  v[t]  above the line  v = -32 t?

h[9.815]v[9.815]

These numerical computations can be modified to find the position of the bungee diver as a function of time when you add the bungee cord.

See HINT 25.2

Find the times when the various divers tighten the cords of lengths 120, 240, 415, and 595.  Add the appropriate computations to the notebook.

The Cord Tightens

After you have entered the computations of  h[t]  and  v[t]  for falling in air, use them to find the time when the various lengths of cord tighten.

FindRoot[h[t] == 120, {t, 5}]

Plot[{h[t], 120}, {t, 0, 9.09423}, PlotStyle -> {Blue, Brown}];

v[9.09423]

So the diver reaches 120 feet above the canyon floor in t=?? seconds.  The velocity at this time is  v[??]=???  Is this when the 120 foot cord tightens?

What are the times and velocities when length = 120, length = 240, length = 415,

The Force Due to the Cord

You are on your own now with explanations.  You need to add variables and parameters to describe the effect of the cord.  Read the printed project and HINTS.

The cord produces no force until it is stretched beyond its naturla length.  Here is a function like that:

sample[x_] := If[x > 2 , x - 1, 1];
Plot[sample[x],{x,0,4},PlotRange -> {0,3}];

HINT 25.5

Use this idea to define the force due to the cord as a function of the

\(Clear[c, length, s]; 
length\  = \ 120; 
s\  = \ 3.4\ 120/length; 

    F\_c[h_] := 
      \(If[h\  < \ \((1053\  - \ length)\)\ , \ 
        \(?? \(\(?
            your\ formula???\ ,\ 0];

              Plot[F\_c[h], \ {h, 0, 1053}, 
                AxesLabel -> {"h", \*""\!\(F\_c\)""}]\); \)\)\)\)

HINT 25.7

What stretch of the various cords produce forces of 160 pounds (or 140 pounds for Krazy Keith and his 595 ft cord)?

Jumping with Air and Cord

Modify the sample solution above for diving in air without a cord to solve for the bungee diver's height and velocity as functions of time.  Either start the solution at the time when the cord begins to stretch and use the velocity at that time, or use the more complicated piecewise formula for cord force and start at the top.

Derivation of the Bungee Equations

Derive the equations of motion for a bungee diver in terms of your variables and parameters.

Clear[t, v, h, m, k, p, F, Subscript, r, c, length, s];
g = 32;
\(\(m\  = \ 160/g;\)
\)
(Subscript[F, r])[v_] := -k Sign[v] Abs[v]^p;
p = (7/5);
\(\(k = 0.10;\)
\)
\(F\_c[h_] := \(?? 
length\ =\ 120;
s\  = \ 3.4*120/length; 
		

        tFinal = 50.0; 


        soln = NDSolve[
		{h[
                  0] == \(?? ,
			v[0] == \(?? ,
			\(h'\)[t] == 
                        v[t], 
			\(v'\)[
                          t] == \(-g\) + \(F\_c[h[t]] + F\_r[v[t]]\)\/m\)\)}, {h[t], v[t]}, {t, 0, tFinal}]; 
h[t_] = h[t] /. soln[\([1, 1]\)]; 

        v[t_] = v[t] /. soln[\([1, 2]\)]; 

        Plot[h[t], {t, 0, tFinal}, AxesLabel -> {"t", "h"}, 
          PlotStyle -> Blue, PlotRange -> {0, 1055}, AxesOrigin -> {0, 0}]; 

        h[tFinal]\)\)

Plot[v[t], {t, 0, tFinal}, AxesLabel -> {"t", "v"}, 
      PlotStyle -> Red, PlotRange -> {-100, 100}, AxesOrigin -> {0, 0}];

You are on your own now with explanations.  Read the printed project and HINTS.

The Big Questions: HINT 25.9

Which divers lived?