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Impulse and Time-Averaged Force     

Impulse and force are closely related. In fact, if a time interval of interest is specified, the impulse imparted by a specific force during that interval can be used to quickly estimate the time-average of that force. The mathematical definition of the time-average of a force is:

\begin{large}\[ \langle\vec{F}\rangle_{t} \equiv \frac{\int_{t_{i}}^{t_{f}}\vec{F}\:dt}{t_{f}-t_{i}}  \]\end{large}

Using the definition of impulse, this expression can be written:

\begin{large}\[ \langle\vec{F}\rangle_{t} = \frac{\vec{J}}{t_{f}-t_{i}}\]\end{large}