Skip to main content
Physics LibreTexts

2.2: Newton's Laws of motion

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    Newton defined a vector quantity called linear momentum \(\mathbf{p}\) which is the product of mass and velocity.

    \[\label{eq:2.1} \mathbf{p} = m\dot{\mathbf{r}}\]

    Since the mass \(m\) is a scalar quantity, then the velocity vector \(\dot{r}\) and the linear momentum vector \(\mathbf{p}\) are colinear.

    Newton’s laws, expressed in terms of linear momentum, are:

    1. Law of inertia: A body remains at rest or in uniform motion unless acted upon by a force.
    2. Equation of motion: A body acted upon by a force moves in such a manner that the time rate of change of momentum equals the force.\[\label{eq:2.2}\mathbf{F} = \frac{d\mathbf{p}}{dt}\]
    3. Action and reaction: If two bodies exert forces on each other these forces are equal in magnitude and opposite in direction.

    Newton’s second law contains the essential physics relating the force \(\mathbf{F}\) and the rate of change of linear momentum \(\mathbf{p}\).

    Newton’s first law, the law of inertia, is a special case of Newton’s second law in that if


    then \(\mathbf{p}\) is a constant of motion.

    Newton’s third law also can be interpreted as a statement of the conservation of momentum, that is, for a two particle system with no external forces acting,

    \[\label{eq:2.4} \mathbf{F}_{12} = -\mathbf{F}_{21}\]

    If the forces acting on two bodies are their mutual action and reaction, then Equation \ref{eq:2.4} simplifies to

    \[\label{eq:2.5} \mathbf{F}_{12}=-\mathbf{F}_{21}= \frac{d\mathbf{p_1}}{dt}+ \frac{d\mathbf{p_2}}{dt} = \frac{d}{dt}(\mathbf{p_1+p_2}) = 0\]

    This implies that the total linear momentum \(\mathbf{P = p_1 + p_2}\) is a constant of motion. Combining Equations \ref{eq:2.1} and \ref{eq:2.2} leads to a second-order differential equation

    \[\label{2.6} \mathbf{F}=\frac{d\mathbf{p}}{dt}=m\frac{d^2\mathbf{r}}{dt^2}=m\mathbf{\ddot{r}}\]

    Note that the force on a body \(\mathbf{F}\), and the resultant acceleration \({\bf a = \ddot{r}}\) are colinear. Appendix \(19.3.2\) gives explicit expressions for the acceleration \({\bf a}\) in cartesian and curvilinear coordinate systems. The definition of force depends on the definition of the mass \(m\). Newton’s laws of motion are obeyed to a high precision for velocities much less than the velocity of light. For example, recent experiments have shown they are obeyed with an error in the acceleration of \(\Delta a \leq 5 \times 10^{-14}\mathit{m/s^2}\).

    This page titled 2.2: Newton's Laws of motion is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Douglas Cline via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.