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Composition Setup
 Excerpt
hiddentrue
Bungee jumps involve elastic and gravitational potential energy. 
Image Added
(Photo courtesy Wikimedia Commons, uploaded by user Che010.)
Bungee cords designed to U.S. Military specifications (DoD standard 
...
 MIL-C-5651D, 
...
 available 
...
 at 
...
http://dodssp.daps.dla.mil
...
) 
...
 are 
...
 characterized 
...
 by 
...
 a 
...
 force 
...
 constant 
...
 times 
...
 unstretched 
...
 length 
...
 in 
...
 the 
...
 range 
...
kL
...
 ~ 
...
 800-1500 
...
 N. 
...
 Jumpers 
...
 using 
...
 these 
...
 cords 
...
 intertwine 
...
 three 
...
 to 
...
 five 
...
 cords 
...
 to 
...
 make 
...
 a 
...
 thick 
...
 rope 
...
 that 
...
 is 
...
 strong 
...
 enough 
...
 to 
...
 withstand 
...
 the 
...
 forces 
...
 of 
...
 the 
...
 jump. 
...
 Suppose 
...
 that 
...
 you 
...
 are 
...
 designing 
...
 a 
...
 bungee 
...
 jump 
...
 off 
...
 of 
...
 a 
...
 bridge 
...
 that 
...
 is 
...
 50.0 
...
 m 
...
 above 
...
 the 
...
 surface 
...
 of 
...
 a 
...
 river 
...
 running 
...
 below. 
...
 You 
...
 have 
...
 read 
...
 that 
...
 you 
...
 should 
...
 expect 
...
 the 
...
 cord 
...
 to 
...
 stretch 
...
 (at 
...
 peak 
...
 extension) 
...
 to 
...
 about 
...
 210% 
...
 of 
...
 its 
...
 natural 
...
 length. 
...
 You 
...
 have 
...
 also 
...
 read 
...
 that 
...
 you 
...
 should 
...
 use 
...
 3 
...
 cords 
...
 together 
...
 for 
...
 jumpers 
...
 with 
...
 weights 
...
 in 
...
 the 
...
 range 
...
 100-150 
...
 lbs, 
...
 4 
...
 cords 
...
 for 
...
 150-200 
...
 lbs, 
...
 and 
...
 5 
...
 cords 
...
 for 
...
 200-250 
...
 lbs. 
...
 Suppose 
...
 you 
...
 decide 
...
 to 
...
 use 
...
 cords 
...
 of 
...
 length 
...
 20 
...
 m, 
...
 which 
...
 would 
...
 seem 
...
 to 
...
 offer 
...
 a 
...
 safety 
...
 zone 
...
 of 
...
 about 
...
 8 
...
 m 
...
 (or 
...
 really 
...
 about 
...
 6 
...
 m 
...
 if 
...
 the 
...
 cord 
...
 is 
...
 attached 
...
 at 
...
 the 
...
 ankles).


{:=}
{
 Deck of Cards
id
partdeck
:= A}

h3. Part A

Find the expected maximum length of the cords for a 200 lb person jumping with 4 cords, so that _kL_ = 
 Card
label
Part 
A
Part A
Find the expected maximum length of the cords for a 200 lb person jumping with 4 cords, so that kL = (4)(800 
...
 N) 
...
 = 
...
 3200 
...
 N. 
...
 Since 
...
 you 
...
 are 
...
 evaluating 
...
 the 
...
 safety 
...
 factor, 
...
 ignore 
...
 any 
...
 losses 
...
 due 
...
 to 
...
 air 
...
 resistance 
...
 or 
...
 dissipation 
...
 in 
...
 the 
...
 cord. 
...
 Ignore 
...
 the 
...
 mass 
...
 of 
...
 the 
...
 rope.
...
Solution
 Toggle Cloak
...
id
...
sysA
...
System:
...
 Cloak
...
id
...
sysA
...
 The 
...
 jumper 
...
 (treated 
...
 as 
...
 a 
...
point 
...
 particle
...
) 
...
 plus 
...
 the 
...
 earth 
...
 (treated 
...
 as 
...
 an 
...
 object 
...
 of 
...
 infinite 
...
 mass) 
...
 and 
...
 the 
...
 bungee 
...
 cord 
...
 and 
...
 the 
...
 bridge 
...
 (also 
...
 treated 
...
 as 
...
 an 
...
 object 
...
 of 
...
 infinite 
...
 mass).
...
 Cloak
...
sysA
sysA
 Toggle Cloak
...
id
...
intA
...
Interactions:
...
 Cloak
...
id
...
intA
...
 The 
...
system 
...
 constituents
...
 interact 
...
 via 
...
 gravity, 
...
 which 
...
 contributes 
...
 gravitational 
...
potential 
...
 energy
...
 and 
...
 via 
...
 the 
...
elastic restoring 
...
 force
...
 of 
...
 the 
...
 cord 
...
 which 
...
 contributes 
...
 elastic 
...
potential 
...
 energy
...
. 
...
 External 
...
 influences 
...
 are 
...
 assumed 
...
 negligible.
...
 Cloak
...
intA
intA
 Toggle Cloak
...
id
...
modA
...
Model:
...
 Cloak
...
id
...
modA
...
Mechanical 
...
 Energy
...
, External Work, and Internal Non-Conservative 
...
 Work
...
.
...
 Cloak
...
modA
modA
 Toggle Cloak
...
id
...
appA
...
Approach:
...
 Cloak
idappA
We begin with an initial-state 
...
 final-state 
...
diagram and 
...
 the 
...
 corresponding 
...
energy 
...
 bar diagrams: 
Image Added
Image Added
Initial State
Final State
As indicated in the picture, we have chosen the zero point of the height to be the river's surface. With these pictures in mind, we can set up the Law of Change for our model:
 Latex
 diagrams: 

|!bungee1a.png!|!bungee1b variant.png!
||Initial State||Final State||

As indicated in the picture, we have chosen the zero point of the height to be the river's surface. With these pictures in mind, we can set up the Law of Change for our model:
{latex}\begin{large} \[ E_{\rm i} = mgh_{\rm i} = E_{\rm f} = mgh_{\rm f} + \frac{1}{2}k x_{f}^{2} \] \end{large}{latex}{note}Bungee cords provide a restoring force when stretched, but offer no resistance when 
 Note
Bungee cords provide an elastic restoring force when stretched, but offer no resistance when "compressed", 
 
 since 
 
 they 
 
 fold 
 
 like 
 
 an 
 
 ordinary 
 
 rope. 
 
 Thus, 
 
 the 
 
 initial 
 
spring 
 
 energy
 
 is 
 
 zero 
 
 in 
 
 this 
 
 case. 
 {note}

It 
...
 seems 
...
 that 
...
 we 
...
 have 
...
 a 
...
 problem 
...
 here, 
...
 because 
...
 we 
...
 do 
...
 not 
...
 know hf or xf. The easiest way to deal with this problem is to utilize our freedom to choose the zero point of the height axis. If we restructure our coordinate system to place h = 0 m at the point where the cord is stretched to its natural length L (as shown below).
Image Added
Image Added
Initial State
Final State
This redefinition of the origin of the height in our coordinate system greatly simplifies the equation describing the evolution of the jumper's mechanical energy:
 Latex
 _h{_}{~}f~ or _x{_}{~}f~. The easiest way to deal with this problem is to utilize our freedom to _choose_ the zero point of the height axis. If we restructure our coordinate system to place _h_ = 0 m at the point where the cord is stretched to its natural length _L_ (as shown below) then we can rewrite our equation: {warning}Note that we are assuming here that the person is much shorter than the cord, and so we will treat the person as a point particle. {warning}

|!bungee2a.png!|!bungee2b variant.png!
||Initial State||Final State||

{latex}\begin{large} \[ mgL = - mgx_{f} + \frac{1}{2} k x_{f}^{2} \] \end{large}{latex}{note}You can also solve using the initial coordinate system. You would simply have to substitute _h{_}{~}f~ = 50 m - _L_ \- _x_. After cancelling _mg_(50 m) from each side, you recover the same expression. {note}

The discussion around the diagrammatic representation has left us with a quadratic equation in _x_~f~, which is solved to obtain:
{latex}
 Note
You can also solve using the initial coordinate system. You would simply have to substitute hf = 50 m - L - x. After cancelling mg(50 m) from each side, you recover the same expression. 
The discussion around the diagrammatic representation has left us with a quadratic equation in xf, which is solved to obtain:
 Latex
\begin{large} \[ x_{f} = \frac{mg \pm \sqrt{(mg)^{2} + 2 k mg L}}{k} \]\end{large}{latex}

It 
...
 is 
...
 not 
...
 sensible 
...
 that 
...
 we 
...
 should 
...
 find 
...
 a 
...
 negative 
...
 value 
...
 for xf, so we must select the plus sign, giving:
 Latex
 _x{_}{~}f~, so we must select the plus sign, giving:
{latex}\begin{large} \[ x_{f} = \frac{mg}{k}\left(1+\sqrt{1+\frac{2kL}{mg}}\right) = 21.5\:{\rm m} \] \end{large}{latex}

{tip}Does this bear out the original estimate that the cord should stretch to 210% of its initial length? {tip}

{tip}Can you convince yourself by looking at the form of the symbolic answer that a 150 lb person attached to 3 cords and a 250 lb person attached to 5 cords would have the same _x{_}{~}f~ as the 200 lb person on 4 cords? {tip}

{cloak:appA}

{card:Part A}
{card:label=Part B}

h3. Part B

Based on your analysis from Part A, what would be the peak force acting on a 200 lb person making the jump attached to 4 cords?

{card:Part B}
{deck:partdeck}

{td}
{tr}
{table}
{live-template:RELATE license}
 Tip
Does this bear out the original estimate that the cord should stretch to 210% of its initial length? 
 Tip
Can you convince yourself by looking at the form of the symbolic answer that a 150 lb person attached to 3 cords and a 250 lb person attached to 5 cords would have the same xf as the 200 lb person on 4 cords? 
 Cloak
appA
appA
 Card
Part A
Part A
 Card
labelPart B
Part B
Based on your analysis from Part A, what would be the peak force acting on a 200 lb person making the jump attached to 4 cords?
 Card
Part B
Part B
 Deck of Cards
partdeck
partdeck