Displacement of Particle under Force

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Let $P$ be a particle of constant mass $m$.

Let the position of $P$ at time $t$ be specified by the position vector $\mathbf r$.

Let a force applied to $P$ be represented by the vector $\mathbf F$.

Then the motion of $P$ can be given by the differential equation:

$\mathbf F = m \dfrac {\d^2 \mathbf r} {\d t^2}$

or using Newtonian notation:

$\mathbf F = m \ddot {\mathbf r}$


\(\ds \mathbf F\) \(=\) \(\ds \map {\dfrac \d {\d t} } {m \mathbf v}\) Newton's Second Law of Motion
\(\ds \) \(=\) \(\ds \map {\dfrac \d {\d t} } {m \dfrac {\d \mathbf r} {\d t} }\) Definition of Velocity
\(\ds \) \(=\) \(\ds m \map {\dfrac \d {\d t} } {\dfrac {\d \mathbf r} {\d t} }\) Derivative of Constant Multiple
\(\ds \) \(=\) \(\ds m \dfrac {\d^2 \mathbf r} {\d t^2}\) Definition of Second Derivative