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ME580
Nonlinear Systems
Due: Sept. 23, 2010
Home work #2
1.
The velocity
v
(
t
)
of a skydiver falling to the ground is governed by
2
mv
mg kv
,
where
m
is the mass of the skydiver,
g
is the acceleration due to gravity, and
k
0
is a
constant related to the air resistance the diver faces.
a) Find the analytical solution for the skydiver's velocity
v
(
t
)
as a function of time,
assuming that the initial velocity is
v
(0)
0.
b) Find the limiting value of
v
(
t
)
as
t
.
(Strogatz: Nonlinear Dynamics and Chaos; pp. 38)
2.
Consider the model chemical reaction
A
X
2
X
,
in which one molecule of X combines with another molecule of A to form two molecules of X.
This implies that the chemical X stimulates its own production, a process called
autocatalysis.
This positive feedback process results in a chain reaction that eventually is limited by a "back
reaction" in which 2X returns to A+X.
According to the
law of mass action
, the rate of an elementary reaction is proportional to
the product of the concentrations of the reactants.
Let
x
and
a
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