Homework 5, Problem Number 8
A woman bails out of an airplane at an altitude of ten thousand feet, falls
freely for twenty seconds , then opens her parachute. Assume linear air resistance v ft/s2 , taking = 0:15 without the parachute and = 1:5 with
the parachute. Suggestion: First determine her height above the ground and
velocity when the parachute opens.
First, observe that the woman's velocity must satisfy the di erential equation
dv
= g v:
dt
(1)
This is a rst order linear di erential equation. Therefore, by writing
dv
+ v = g
(2)
dt
we see that we have P (t) = and Q(t) = g. Multiplying both sides of this
equation by the integrating factor
(t) = exp
yields
et
Equivelantly, this gives
Z
dt = et ;
dv
+ et v = get :
dt
et v 0 = get :
Therefore, we see that
e v= g
t
and
Z
v(t) =
et dt =
g
+ Ce
(3)
(4)
(5)
g t
e + C;
t
(6)
(7)
is the general solution to (1).
To determine the woman's velocity when the parachute opens, we need to
solve for C using the initial condition v(0) = 0.
This gives,
0 = g + C;
(8)
so the woman's velocity at time t before she opens her parachute is given by
v(t) =
g g
+ e
where g = 32 and = 0:15.
1
t
;
(9)
Therefore, at time t = 20, her velocity is v(20) = 202:712 feet per second.
Notice that in general we will have
g
C = v(0) + :
(10)
To nd her height above the ground when the parachute opens, rst notice
that
y(t) =
Z
Z
g g t
+ e dt + C1
= g t g2 e t + C1 :
v(t)dt + C1 =
(11)
Because y(0) = 10000, we see that
10000 = g2 + C1 ;
so the woman's height above the ground at time t is given by
y(t) =
g
g
t+ e
2
t
+ 10000 + g2
(12)
(13)
Therefore, at time t = 20, her height is y(20) = 7084:75 feet. Notice that in
general we will have
v(0) g
C1 = y(0) +
(14)
+ 2
To determine how long it will take her to reach the ground, we will need
to solve a very similar problem, using the velocity and height just computed as
new initial conditions. Therefore, the woman's velocity at time t seconds after
opening her parachute is the solution to the initial value problem
dv
= g v; v(0) = 202:712:
dt
(15)
Just as before, the general solution to this di erential equation is given by (7).
Therefore,
g
g
(16)
v(t) = t + v(0) + e t :
Similarly, her position is given by
v(0) g
g
e
+
y(t) = t
2
= 21:3333t + 120:919e
t
g
+ y(0) + v(0)
+
2
1 :5 t
+ 6964:83
Setting y(t) = 0 and solving for t (numerically) gives t = 326:476.
2
(17)