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 Excerpt
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hidden
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true
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A 
...
 more 
...
 complicated 
...
 version 
...
 of 
...
 the 
...
Bungee 
...
 Jump
...
 problem. 
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 
...
 30.0 
...
 m 
...
 above 
...
 the 
...
 surface 
...
 of 
...
 a 
...
 river 
...
 running 
...
 below. 
...
 To 
...
 get 
...
 an 
...
 idea 
...
 for 
...
 the 
...
 maximum 
...
 cord 
...
 length, 
...
 calculate 
...
 the 
...
 unstretched 
...
 length 
...
 of 
...
 cord 
...
 with 
...
kL
...
 = 
...
 4000 
...
 N 
...
 (a 
...
 system 
...
 of 
...
 five 
...
 800 
...
 N 
...
 cords) 
...
 that 
...
 will 
...
 result 
...
 in 
...
 a 
...
 118 
...
 kg 
...
 jumper 
...
 who 
...
 leaves 
...
 the 
...
 bridge 
...
 from 
...
 rest 
...
 ending 
...
 their 
...
 jump 
...
 2.0 
...
 m 
...
 above 
...
 the 
...
 water's 
...
 surface. 
...
 (The 
...
 2.0 
...
 m 
...
 builds 
...
 in 
...
 a 
...
 safety 
...
 margin 
...
 for 
...
 the 
...
 height 
...
 of 
...
 the 
...
 jumper, 
...
 so 
...
 you 
...
 can 
...
 neglect 
...
 the 
...
 height 
...
 of 
...
 the 
...
 jumper 
...
 in 
...
 your 
...
 calculation.) 
...
 Since 
...
 you 
...
 are 
...
 finding 
...
 a 
...
 maximum 
...
 cord 
...
 length, 
...
 ignore 
...
 any 
...
 losses 
...
 due 
...
 to 
...
 air 
...
 resistance 
...
 or 
...
 dissipation 
...
 in 
...
 the 
...
 cord. 
...
 Ignore 
...
 the 
...
 mass 
...
 of 
...
 the 
...
 rope.
...
Solution
 Toggle Cloak
...
id
...
sys
...
System:
...
 Cloak
...
id
...
sys
...
The 
...
 jumper 
...
 (treated 
...
 as 
...
 a 
...
 ) 
...
 plus 
...
 the 
...
 earth 
...
 and 
...
 the 
...
 bungee 
...
 cord.
...
 Toggle Cloak
...
id
...
int
...
Interactions:
...
 Cloak
...
id
...
int
...
The 
...
 system 
...
 constituents 
...
 interact 
...
 via 
...
 gravity, 
...
 which 
...
 contributes 
...
gravitational 
...
 potential 
...
 energy
...
, 
...
 and 
...
 via 
...
 the 
...
 restoring 
...
 force 
...
 of 
...
 the 
...
 cord, 
...
 which 
...
 contributes 
...
elastic 
...
 potential 
...
 energy
...
. 
...
 External 
...
 influences 
...
 are 
...
 assumed 
...
 negligible.
...
 Toggle Cloak
...
id
...
mod
...
Model:
...
 Cloak
...
id
...
mod
.
 Toggle Cloak
...
id
...
app
...
 Approach:
*
{:=}
{
 Cloak
id
app
 Toggle Cloak
...
iddiag
 Diagrammatic Representation
 Cloak
iddiag
We begin with an initial-state final-state diagram for this situation, along with corresponding energy bar graphs. 
Image Added
Image Added
Initial State
Final State
 Cloak
diag
diag
 Toggle Cloak
idmath
 Mathematical Representation
 Cloak
idmath
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
=diag} {color:red}{*}Diagrammatic Representation{*}{color}
{cloak:id=diag}
We begin with an initial-state final-state diagram for this situation, along with corresponding energy bar graphs. {table}{tr}{td:valign=bottom} !Bungee Jump^bungee1a.jpg! {td}{td:valign=bottom} !Bungee Jump^bungee1b variant.png! {td}{tr}{tr}{th:align=center}Initial State {th}{th:align=center}Final State {th}{tr}{table}
{cloak:diag}
{toggle-cloak:id=math} {color:red}{*}Mathematical Representation{*}{color}
{cloak:id=math}
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}{info}Bungee cords provide a restoring force when stretched, but offer no resistance when 
 Info
Bungee cords provide a 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. 
 {info}

This 
...
 equation 
...
 cannot 
...
 be 
...
 solved 
...
 without 
...
 further 
...
 constraints, 
...
 since 
...
 we 
...
 do 
...
 not 
...
 know 
...
k
...
. 
...
 The 
...
 extra 
...
 constraint 
...
 that 
...
 we 
...
 have 
...
 is 
...
 given 
...
 by 
...
 the 
...
 fact 
...
 that 
...
 the 
...
 jumper 
...
 has 
...
 fallen 
...
 a 
...
 total 
...
 of 
...
 28.0 
...
 m 
...
 (descending 
...
 from 
...
 30.0 
...
 m 
...
 above 
...
 the 
...
 water 
...
 down 
...
 to 
...
 2.0 
...
 m 
...
 above 
...
 the 
...
 water). 
...
 This 
...
 distance 
...
 must 
...
 be 
...
 covered 
...
 by 
...
 the 
...
 stretched 
...
 cord. 
...
 This 
...
 gives 
...
 us 
...
 the 
...
 contraint:

{
 Latex
}\begin{large} \[ h_{\rm i} - h_{\rm f} = L + x_{f} \] \end{large}{latex}

Solving 
...
 this 
...
 constraint 
...
 for xf and substituting into the energy equation gives:
 Latex
 _x{_}{~}f~ and substituting into the energy equation gives:
{latex}\begin{large} \[ -2mg(\Delta h) = k((\Delta h)^{2} + 2 L \Delta h + L^{2}) \] \end{large}{latex}

Where 
...
 we 
...
 are 
...
 using 
...
 Δ
...
h
...
 = hf - hi to simplify the expression.
 We still have two unknowns, but we can resolve this by using the fact that kL is a constant for the rope. Thus, if we multiply both sides by L, we have:
 Latex
 _h{_}{~}f~ \- _h{_}{~}i~ to simplify the expression.
We still have two unknowns, but we can resolve this by using the fact that _kL_ is a constant for the rope. Thus, if we multiply both sides by _L_, we have:
{latex}\begin{large} \[ -2mg(\Delta h)L = C((\Delta h)^{2} + 2L \Delta h + L^{2}) \] \end{large}{latex}

where 
...
 we 
...
 have 
...
 replaced 
...
 the 
...
 quantity 
...
kL
...
 by 
...
 C 
...
 (= 
...
 800 
...
 N) 
...
 for 
...
 clarity. 
...
 With 
...
 this 
...
 substitution, 
...
 it 
...
 can 
...
 be 
...
 seen 
...
 that 
...
 we 
...
 have 
...
 a 
...
 quadratic 
...
 equation 
...
 in 
...
L
...
 which 
...
 can 
...
 be 
...
 solved 
...
 to 
...
 find:

{
 Latex
}\begin{large} \[ L = \frac{-2(C+mg)\Delta h \pm \sqrt{4(C+mg)^{2} (\Delta h)^{2}-4C^{2}(\Delta h)^{2}}}{2C} = 13.3 \:{\rm m}\;{\rm or}\;58.9 \:{\rm m}\] \end{large}{latex}

It 
...
 is 
...
 clear 
...
 that 
...
 the 
...
 appropriate 
...
 choice 
...
 is 
...
 13.3 
...
 m. 
 {
 Note
}
Does 
 
 the 
 
 other 
 
 root 
 
 have 
 
 a 
 
 physical 
 
 meaning? 
 Tip
titleA Rule of Thumb
According to bungee jumping enthusiasts, a good rule of thumb is to expect that the maximum length reached by a bungee cord during a jump will be 210% of its unstretched length (assuming the cord is rated for the weight of the person jumping). Thus, a 10 m cord should stretch to 21 m during a jump. How does this rule of thumb compare to our calculated estimate? 
Follow Up – Checking Assumptions
In our constraint that:
 Latex
 {note}{tip:title=A Rule of Thumb}According to bungee jumping enthusiasts, a good rule of thumb is to expect that the maximum length reached by a bungee cord during a jump will be 210% of its unstretched length (assuming the cord is rated for the weight of the person jumping). Thus, a 10 m cord should stretch to 21 m during a jump. How does this rule of thumb compare to our calculated estimate? {tip}
{cloak:math}
{cloak:app}

----
h3. Follow Up -- Checking Assumptions

In our constraint that:
{latex}\begin{large}\[ h_{i} - h_{f} = L + x_{f} \] \end{large}{latex}

we 
...
 neglected 
...
 the 
...
 fact 
...
 that 
...
 the 
...
 jumper's 
...
 center 
...
 of 
...
 mass 
...
 might 
...
 drop 
...
 an 
...
 extra 
...
 meter 
...
 or 
...
 more 
...
 (depending 
...
 upon 
...
 the 
...
 point 
...
 of 
...
 attachment 
...
 of 
...
 the 
...
 cord 
...
 to 
...
 the 
...
 person). 
...
 What 
...
 effect 
...
 would 
...
 such 
...
 an 
...
 extra 
...
 drop 
...
 have 
...
 on 
...
 the 
...
 final 
...
 height?
Follow Up – Checking Assumptions
Suppose a 118 kg jumper foiled your calculations by leaping upward from the bridge with an initial speed of 2.25 m/s. Will the jumper hit the water, assuming the 13.3 m cord with kL = 4000 N that we found in Part A?
 Cloak
math
math


----
h3. Follow Up -- Checking Assumptions

Suppose a 118 kg jumper foiled your calculations by leaping upward from the bridge with an initial speed of 2.25 m/s. Will the jumper hit the water, assuming the 13.3 m cord with _kL_ = 4000 N that we found in Part A?
 Cloak
app
app