Numerically integrating equations of motion
Newtons equations give an equation for the acceleration of an object.
~a
=
~
F
m
If you happen to know
~
F
as a function of time, one can Fnd position as a
function of time by integrating twice:
~v
(
t
)=
~v
(0) +
!
t
0
~
F
(
¯
t
)
m
d
¯
t
~r
(
t
~r
(0) +
!
t
0
~v
(
¯
t
)
d
¯
t.
These equations just mean that you draw the graph of each component of
(
t
)
and then Fnd the areas to get
~v
(
t
). You then repeat to get
~r
(
t
).
Life is usually more complicated. In most realworld applications the force is
not given to you as a function of
t
, but instead the force depends on where the
object is [or even how fast its going]. ±or example, specializing to one dimension,
an object attached to a spring will experience a force
F
=
−
kx,
giving an equation of motion
d
2
x
(
t
)
dt
2
=
−
k
m
x
(
t
)
.
Since this equation involves a derivative it is known as a “di²erential equa
tion”. It turns out that this simple example can be solved exactly, however at
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '07
 LECLAIR,A
 mechanics, Derivative, Acceleration, Numerics, Numerically integrating equations

Click to edit the document details