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h3. The Integral Form of Newton's Second Law and 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 [force] explicit in this equation, but in practice it is not useful.  If a known [force] 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 [impulse] 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 [Law of Change] 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}