The Parachute Problem
(adapted from Douglas B. Meade’s
“ODE Models for the Parachute Problem”)
Noam Goldberg
Craig Kaplan
Tucker Riley
Outline
Introduction to the Parachute Problem
 Modeling the Descent
 Derivation of Equations
 Application of Equations
 Results

The Parachute Problem
Newton’s
Second Law:
F  ma
x
x0
v
Figure 1: Forces acting on a skydiver in flight
Mathematical Model of Falling
Object
m x' '  m g  kx'
Or, in the case of the Parachute Problem:
 m g  k x '
1

m x"  
 m g  k 2 x '

t  td
t  td
Equivalent First-Order System
k
v '   g  v,
m
v(0)  0,
x '  v,
x ( 0)  x 0 ,
Solving for Velocity v'   g  k v
m
k
k
v'   g  v → v' v   g
m
m
k
↓
v' v   g
v' P(t )v  Q(t )
m
(We can use an integrating factor!)
k
P(t )  ,
m
Q(t )   g
Integrating Factor
P(t ) dt


 e
e
v
e
k
dt
m
1
Q(t ) dt


k
 t
m
 (  g )e
k
 t
m
  ge
k
 t
e

k
t
m
k
t
m
dt
dt
k
t

k
t
m
e .
k
 t
m
gm
0
Finding the
Integrating
Constant

e  gAe
k
k
 t (General
m velocity
mg
v(t )  
 gAe
k
Using the initial condition:
v(0)  0
We can solve for A:
m
A
k
.
equation)
(2.15
Velocity Equation Before Deployment
mg
e
v(t ) 

k1 
k1
 t
m

 1,

t  td
Solving for Position
x (t )   v (t ) dt
k u-substitution,
Using our velocity
equation,

t


mg
m
and
 e condition:
 the initial
 1dt
x(0k) x ,

0

k
we seethat

mg   m t
 2 e
 1dt. k
  m t 
m kg  k
x(t )  x0  2  t  1  e 
k
 m


Solving for Velocity After Deployment
At t=td we have a new initial condition:
v(td )  v0
Plugging this value into the general velocity
equation (the one we had before plugging in
ICs), we obtain:
k
 td
mg
v(td )  v0  
 gAe m
k
Solving for A, we get:
A  e
k
td
m
 m v0 
  
k g
Solving for Velocity After Deployment

Plugging A into the original velocity equation, we find
that:
k
k
2
td t 
mg mg m2 td t 
v(t )  

e
em
v0
k2
k2
To get v0, we plug t=td into our equation for
velocity before deployment:
k1


m g m td 
v 0  v(td ) 
1
e
k1 

Solving for Velocity After Deployment
k1

td


mg mg
mg
 e m  1
v(t )  

e
e
 k

k2
k2 k2
k2
k 2 k1 

td  t m g   1 td
m g m g m t d  t 
m
m
m

v(t )  

e

1
k1
k 2↓  e
k 2 e
m gk2  m tkd 2   m t td  m g  k1 mt td   
e
e
e

v(t ) 

1


1
 k
 k

k1  k1
k
2   2 t t 
2



t


t

t



mg
m
g
d
d
d
 e m  1e m
e m

v(t ) 


1



k1 
k
2 


k2
t d  t 
m
k2
k
td  2 t
m
m
Velocity Equation for Whole Jump
 m g   k1 t 
 e m  1



k
1



v(t )  
k1
k2
k2

t


t

t


t  t d 




m
g
m
g
d
d

m
m
m

e
 1e

 1
 k e

k2 


 1 
t  td
t  td
Applications of Derived Equations
h = 3500m
td = 60 seconds
m
k2 5
k1 1
g  9 .8 2


s
m 3
m 6
1. When the chute is pulled, what is the elevation
and velocity of the skydiver?
2. How long is the total jump?
3. At what velocity does the skydiver hit the
ground?
1. Elevation and velocity at time of deployment
1
 ( 60 )  

36  9.8  1
x(60)  3500
 (60)  1  e 6 
1  6


x60  325m
1


6  9.8  6 ( 60 ) 
v(60) 
e
 1

1 

v60  58.8 m/s
2. Total length of jump
Setting x(tf) = 0, solve for tf
5
 tf

9  9.8  5
x(t f )  0  324.784
 t f  1  e 3
25  3

t f  55.835 60  116 s




3. Final velocity
Solve for v(116)
1
5
5


60


116

60


116 60 




6  9.8
3  9.8
6
3
3
e
e
v(116) 
 1e

 1


1 
5 


v(116)  5.88 m/s
Graph of negative velocity versus time
-velocity (m/s)
time (s)
Applications of Derived Equations
For
h = 3500m
td = 60 seconds
k1 1

m 6
k2 5

m 3
g  9 .8
m
s2
x(td) = 325m, v(td) = -58.8 m/s ≈ 130 mi/hr
2. tf = 116 s
3. v(tf) = -5.88 m/s
1.
Conclusion
Verified that Meade’s equations are correct by
deriving them ourselves
 Used derived equations to find various
velocities and positions, and total time of a
typical jump (Meade)

FOR SALE:
PARACHUTE
ONLY USED ONCE
NEVER OPENED
SMALL STAIN
Works Cited
Blanchard, Paul, Robert L. Devaney, and
Glen R. Hall. Differential Equations.
Third Edition. Belmont, CA: Brooks/Cole,
2006. Print.
 Meade, Douglas B. “ODE Models for the
Parachute Problem.” Siam Review 40.2
(1998): 327-332. Web. 27 Oct 2010.
