...
The
...
Integral
...
Form
...
of
...
Newton's
...
Second
...
Law
...
and Impulse 
The Law of Change for the Momentum and External Force model can in principle be integrated:
| Latex |
|---|
Impulse [!images^SectionEdit.png!|Momentum (Impulse)] The Law of Change for the [Momentum and External Force] [model] can in principle be integrated: {latex}\begin{large}\[ \int_{\vec{p}_{i}}^{\vec{p}_{f}} d\vec{p} = \int_{t_{i}}^{t_{f}} \sum_{\rm ext} \vec{F}\:dt\]\end{large}{latex} |
The
...
left
...
hand
...
side
...
of
...
this
...
expression
...
is
...
simple,
...
and
...
after
...
some
...
rearrangement,
...
the
...
equation
...
becomes:
| Latex |
|---|
}\begin{large} \[ \vec{p}_{f} = \vec{p}_{i} + \int_{t_{i}}^{t_{f}} \sum_{\rm ext} \vec{F}\:dt\]\end{large}{latex} |
In
...
principle,
...
it
...
might
...
be
...
useful
...
to
...
leave
...
the
...
integral
...
over
...
...
explicit
...
in
...
this
...
equation,
...
but
...
in
...
practice
...
it
...
is
...
not
...
useful.
...
If
...
a
...
known
...
...
which
...
is
...
an
...
easily
...
integrable
...
function
...
of
...
time
...
is
...
applied,
...
then
...
it
...
is
...
usually
...
just
...
as
...
simple
...
and
...
more
...
intuitive
...
to
...
use
...
the
...
traditional
...
F
...
=
...
ma
...
approach
...
(followed
...
by
...
regular
...
kinematics).
...
The
...
utility
...
of
...
this
...
equation
...
actually
...
lies
...
in
...
the
...
reverse
...
approach:
...
using
...
what
...
is
...
known
...
about
...
momentum
...
to
...
learn
...
about
...
the
...
force.
...
To
...
facilitate
...
this,
...
we
...
define
...
the
...
...
associated
...
with
...
a
...
force
...
as:
| Latex |
|---|
}\begin{large}\[ \vec{J} = \int \vec{F}\:dt \]\end{large}{latex} |
With
...
this
...
definition,
...
the
...
integral
...
form
...
of
...
the
...
...
...
...
can
...
be
...
written:
| Latex |
|---|
}\begin{large}\[ \vec{p}_{f} = \vec{p}_{i} + \sum_{\rm ext} \vec{J} \]\end{large}{latex} {panel:bgColor=#F0F0FF}!images^SAP.gif! *[Off the Wall]* ({excerpt-include:Off the Wall|nopanel=true}) {panel} |
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