# 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:

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

or using Newtonian notation:

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

## Proof

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

$\blacksquare$