Impulse null
When expressed in terms of momentum, Newton's Second Law can in principle be integrated:
\begin
[ \int_{\vec
_{i}}^{\vec
_{f}} d\vec
= \int_{t_{i}}^{t_{f}} \sum_
\vec
\:dt]\end
The left hand side of this expression is simple, and after some rearrangement, the equation becomes:
\begin
[ \vec
_
= \vec
_
+ \int_{t_{i}}^{t_{f}} \sum_
\vec
\:dt]\end
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
[ \vec
= \int \vec
\:dt ]\end
With this definition, the integral form of Newton's Second Law can be written:
\begin
[ \vec
_
= \vec
_
+ \sum_
\vec
]\end