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1.5 Moving objects – Newton's 2nd Law.
Another type of dynamical system is a collection of moving objects. The motion of objects is described by
Newton's 2nd law of motion.
Example 1. Suppose we have a falling object. Let the x axis by the vertical line in which the object is
falling. So
x = x(t) = position of object
-1
t = time measured from some starting time
0
1
dx
v =
= velocity = rate of change of position
dt
dv
d2x
= 2 = acceleration = rate of change of velocity = 2nd derivative of position
dt
dt
a =
Newton's 2nd law says the objects mass times its acceleration is the sum of all the forces on the object, i.e.
(1)
m
dv
= F
dt
where
x
F = sum of all the forces on the object
As a specific example, suppose the forces on the object are the force of gravity and the force of air
resistance, i.e.
F = Fg + Fr
where
Fg = force of gravity
Fr = force of air resistance
The force of gravity is the mass times the acceleration of gravity, i.e.
Fg = mg
where
g = acceleration of gravity = 9.8 m/s2
The force of air resistance is more complicated. Let's suppose the force of air resistance is proportional to
the cube root of the velocity and directed opposite to the velocity, i.e.
1.5 - 1
Fr = - k v1/3
where
k = a positive proportionality constant
Combining the above we get
m
dv
= mg - k v1/3
dt
or
dv
k
= g - v1/3
dt
m
This has the form
dv
= f(v)
dt
with
f v
20
k 1/3
v
m
f(v) = g -
15
10
See graph at right.
5
What are the equilibrium points. We must solve f(v) = 0. This is
5
10
5
15
mg 3
20
25
v
k
k
g - v1/3 = 0
m
k 1/3
v
= g
m
So the only equilibrium point is
3
v* =
mg
k
x
We can see from the graph that f(v) > 0 for v < v* and f(v) <
30
0 for v > v*. So if the initial velocity v(0) is less than v*
25
then the object speeds up and the velocity approaches v* as t
20
 . On the other hand, if v(0) is greater than v* then the
15
object slows down and the velocity approaches v* as t  .
10
v* is a sink. At the right are the graphs of some typical
solutions.
5
5
1.5 - 2
10
15
20
t