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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 |
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Second Law] 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
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left
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hand
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side
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of
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this
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expression
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is
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simple,
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and
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after
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some
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rearrangement,
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the
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equation
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becomes:
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}\begin{large} \[ \vec{p}_{f} = \vec{p}_{i} + \int_{t_{i}}^{t_{f}} \sum_{\rm ext} \vec{F}\:dt\]\end{large}{latex} |
In
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principle,
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it
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might
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be
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useful
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to
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leave
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the
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integral
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over
...
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explicit
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in
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this
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equation,
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but
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in
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practice
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it
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is
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not
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useful.
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If
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a
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known
...
...
which
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is
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an
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easily
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integrable
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function
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of
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time
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is
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applied,
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then
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it
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is
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usually
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just
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as
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simple
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and
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more
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intuitive
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to
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use
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the
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traditional
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F
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=
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ma
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approach
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(followed
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by
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regular
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kinematics).
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The
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utility
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of
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this
...
equation
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actually
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lies
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in
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the
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reverse
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approach:
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using
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what
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is
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known
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about
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momentum
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to
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learn
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about
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the
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force.
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To
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facilitate
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this,
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we
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define
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the
...
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associated
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with
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a
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force
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as:
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}\begin{large}\[ \vec{J} = \int \vec{F}\:dt \]\end{large}{latex} |
With
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this
...
definition,
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the
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integral
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form of the Law of Change can be written:
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of [Newton's Second Law] 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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