Linear Acceleration

We begin studying mechanics on an non intertial frame with cases of linear acceleration. To do so imagine an inertial frame \(S_0\) and an accelerating frame \(S\). Relative to \(S_0\), \(S\) is moving at a velocity \(\mathbf{V}\) and acceleration \(\mathbf{A}\). A ball of mass \(m\) is in the \(S\) frame, let’s explore the forces on the ball.

The ball follow’s Newton’s second law in the inertial frame thus,

\[ m\ddot{\mathbf{r}} = \mathbf{F} \tag{Newton's 2nd Law - Inertial Frame} \]

Where \(\dot{\mathbf{r_0}}\) is defined with the Gaussian transform (vector-addition formula) of the ball’s velocity relative to \(S\) and the velocity of \(S\),

\[ \dot{\mathbf{r}} = \dot{\mathbf{r}} + \mathbf{V} \]

We wish to know Newton’s second law for the observer in the accelerating frame \(S\). Therefore we need \(\ddot{\mathbf{r}}\),

\[ \ddot{\mathbf{r}} = \ddot{\mathbf{r}} - \mathbf{A} \]

\[\begin{equation} \boxed{m\ddot{\mathbf{r}} = \mathbf{F} - \underbrace{\hphantom{\ \ }m\mathbf{A}\hphantom{\ \ }}_{\mathbf F_\text{inertial}}}\tag{Newton's 2nd Law - Linear Accelerating Frame} \end{equation}\]

• \(\mathbf{F}_\text{inertial}\) : The non-intertial frame introduces the force called the inertial force or infamously the ficticious force.