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{excerpt}The mathematical relationship between [force] and [momentum], or, for [systems|system] with constant mass, the relationship between [force] and [acceleration].{excerpt}

h4. Motivation for Concept

When you push, kick or use some other means to apply [force] to an object, its [velocity] will change.  It is of value to be able to quantitatively define the strength of such a push, kick or other [force] by examining the effects of the [force] on the object which is the target.  Such a quantitative understanding of [force] is the basis of the science of dynamics.

h4. Statement of the Law
h6. Newton's Statement
"A change in [motion|quantity of motion] is proportional to the motive force impressed and takes place along the straight line in which that force is impressed." (_The Principia_ by I. Newton, translated by I.B. Cohen and A. Whitman.)
h6. Modern Statement
The modern form of the Law which is perhaps most consistent with Newton's statement is the _integral_ formulation:

{latex}\begin{large}\[ \Delta \vec{p} = \int\: \vec{F} \:dt \]\end{large}{latex}

It is more common to express the Law in a differential form:

{latex}\begin{large}\[ \frac{d\vec{p}}{dt} = \vec{F}\]\end{large}{latex}

and, since it is rare to consider a system with changing mass, this form is often reduced to:

{latex}\begin{large}\[ \vec{F} = \frac{d(m\vec{v})}{dt} \rightarrow m\frac{d\vec{v}}{dt} = m\vec{a} \]\end{large}{latex}

h4. Use of the Law in Problem Solving
h6. Form of the Second Law for Multiple Impressed Forces
When more than one force is impressed, the change in momentum is proportional to the [vector] sum of the forces.  Thus, Newton's 2nd Law is usually expanded to state:

{latex}\begin{large}\[ \sum_{j=1}^{N_{F}} \vec{F}_{j} = m\vec{a}\]\end{large}{latex}

h6. "Writing" Newton's Second Law
When a physics teacher or student says they are "writing Newton's 2nd Law" for a system, the form used should be that of the previous subsection, but expressed as _several_ equations separated by vector component:

{latex}\begin{large}\[ \sum_{j=1}^{N_{F}} F_{j,x} = ma_{x}\]
\[\sum_{j=1}^{N_{F}} F_{j,y} = ma_{y} \]
\[\sum_{j=1}^{N_{F}} F_{j,z} = ma_{z} \]\end{large}{latex}

{note}Directions that clearly have no forces acting are sometimes ignored.{note}

h6. Free Body Diagrams
[Free body diagrams|free body diagram] are pictorial guides to the _specific_ form of Newton's 2nd Law for a given system.  Drawing an accurate free body diagram shows visually what forces should be included in the statement of the law, and also gives information about the sign of the vector components of these forces.

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