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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:

\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}

The left hand side of this expression is simple, and after some rearrangement, the equation becomes:

\begin{large} \[ \vec{p}_{f} = \vec{p}_{i} + \int_{t_{i}}^{t_{f}} \sum_{\rm ext} \vec{F}\:dt\]\end{large}

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:

\begin{large}\[ \vec{J} = \int \vec{F}\:dt \]\end{large}

With this definition, the integral form of the Law of Change can be written:

\begin{large}\[ \vec{p}_{f} = \vec{p}_{i} + \sum_{\rm ext} \vec{J} \]\end{large}