Bungee Jump
Copyright 1999
Department of Mathematics
University of Georgia
Athens, Georgia
Kathryn Braungart
Carol W. Penney
Consider a function
that gives the position of an object as a function of time
. This may be the height of a ball
seconds after it has been thrown, the distance a train has moved along a track
hours after it has left the station, or the distance covered by a sprinter after
seconds. In this project you will study the relationship between the position of an object and the velocity of that object at time
.
You know that if an object travels
feet in
seconds, then the
average velocity
of the object from time
to time
is the change of distance,
, divided by the change in time,
. This average velocity is a simple arithmetic calculation, which in this case is described in units of ft/s.
What, then, is the
instantaneous velocity
(what we will simply call
velocity
) of an object at a particular instant? You can calculate average velocities of the object from time
to time
as
. We define the
velocity
of an object at a particular instant
to be the limit of the average velocity from time
to time
as
approaches 0. So if
is the velocity of the object at time
, then
.
Do you recognize this limit? It should look familiar.
In this project you will investigate the relationship between the position of an object and its velocity. In part 1 you will investigate the velocity of a ball being thrown straight up into the air.
Project - Part 1: Velocity of a Ball:
The following movie shows a ball as it rises and falls, along with a graph of the height of the ball above ground level as a function of time. This height function is:
, where
represents the time in seconds after the ball has been thrown. Set the movie to loop, slow down the motion to about 4 frames per second, and view the movie until you understand everything shown in the movie.
![[Maple Plot]](images/bungee25.gif)
Problem 1:
Use the data in the movie to answer the following questions. Write a paragraph describing the motion of the ball; include in your paragraph the answers to these questions and any other observations that you make about the motion of this ball. Show all calculations and present your reasoning clearly. Organize and present your work in a report, rather than simply numbering your solutions to these questions.
represent?
and the velocity of the ball?
-axis from
to
. What does this indicate? What does it have to do with the velocity of the ball?
Problem 2:
Use the function
and your knowledge of functions and derivatives to solve the following problems. Write a paragraph describing the motion of the ball; include in your paragraph your solutions of these problems and of any other problems you investigate about the motion of this ball. Show all calculations and present your reasoning clearly. Organize and present your work in a report, rather than simply numbering your solutions to these questions.
at the point (
).
.
Now you are ready, in Part 2, to investigate A. J. Hackett's record bungee jump:
Project - Part 2: Bungee Jumper
In 1998, A. J. Hackett bungee jumped from Auckland's highest building, setting a world record for a bungee jump from a building. The jump was viewed worldwide over the Internet. The function giving A. J.'s height above ground in feet as a function of time
, where
is measured in seconds, is
.
Investigate A. J.'s jump. In the course of your investigations, make a movie of A. J.'s jump, determine answers to the following questions, then write a paragraph describing his exciting jump. Be sure to show your calculations and present your line of reasoning clearly to the reader. Organize and present your work in a report, rather than simply numbering your solutions to these questions.
1. How close to the ground did A.J. get?
2. How far did he fall on the initial bounce of his breathtaking plunge?
3. How long did it take for A.J. to reach the lowest point of his initial bounce?
4. How high above the ground did A.J. climb on his first bounce?
5. At the conclusion of his jump he was hauled by his bungee cord up to the point from which he jumped. Through what distance was A.J. lifted?
6. How fast was A.J. falling the first time he was 100 feet above ground? The second time he was 100 feet above the ground? When he was 5 feet above ground?
7. What was A. J.'s maximum speed during the jump?
Note: There have been higher bungee jumps, including a 3300 foot jump from a helicopter and a 700 foot jump off a dam in a stunt for the 1995 James Bond movie "Goldeneye".
The following set of commands will generate an animation of A.J.'s jump. You must supply the height function s and the final time in seconds.
| > | restart:with(plots):with(plottools): s:=t->-173.074*exp(-.1*t+.8945)*cos(.38*t-3.39910)+181.602: finaltime:=100: heightfunction:=k->plot([[t,s(t),t=1e-10..k],[t,s(t),t=0..finaltime]], color=[navy,red],thickness=[2,1]): hackett:=k->point([finaltime+5,s(k)],color=black,symbol=circle): bungeecord:=k->line([finaltime+5,s(0)],[finaltime+5,s(k)], color=black): frame:=k->display({ heightfunction(k*finaltime/50), hackett(k*finaltime/50), bungeecord(k*finaltime/50)}, insequence=false): display(seq(frame(k),k=0..50),insequence=true, view=[0..finaltime+10,0..s(0)+10]); |
| > |
The Most Common Maple Commands
Academic Honesty Statement:
Place the following statement (by copying and pasting) at the end of your report and sign it in ink. Your instructor will not grade your report unless this signed statement appears at the end of your report.
I understand that I may work with others if I give them credit in this statement. I also understand that I am required to write my report--that to copy all or part of someone else's report or to allow someone else to copy all or part of my report constitutes plagiarism, which is a serious violation of academic honesty.
I worked with (replace this parenthetical remark with first and last names of those with whom you worked) on this project. I wrote my own report. I did not copy any of this report from anyone else and I did not allow anyone else to copy any of this report.
Signed: