Displacement of Particle under Force

Theorem

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:

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

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

$\ds \mathbf F$

$=$

$\ds \map {\dfrac \d {\d t} } {m \mathbf v}$

$\ds $

$=$

$\ds \map {\dfrac \d {\d t} } {m \dfrac {\d \mathbf r} {\d t} }$

Definition of Velocity

$\ds $

$=$

$\ds m \map {\dfrac \d {\d t} } {\dfrac {\d \mathbf r} {\d t} }$

$\ds $

$=$

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

$\blacksquare$