Tangential and normal components of acceleration

Acceleration split into a part along velocity ( $a_T$ , speeding up/slowing down) and a part perpendicular to it ( $a_N$ , turning).

Acceleration splits into $a_T\mathbf{T}+a_N\mathbf{N}$ , a part parallel to velocity and a part perpendicular to it — the only two directions available, since the $\mathbf{B}$ -coefficient is always exactly $0$ . $a_T=\dfrac{\mathbf{r}'\cdot\mathbf{r}''}{|\mathbf{r}'|}$ changing sign is what "speeding up" versus "slowing down" means in vector terms; $a_N=\dfrac{|\mathbf{r}'\times\mathbf{r}''|}{|\mathbf{r}'|}=\kappa v^2$ is always $\ge0$ and vanishes only where the path is momentarily straight ( $\kappa=0$ ) or momentarily stopped ( $v=0$ ).

Examples

For $\mathbf{r}(t)=\langle t, t^2, 0\rangle$ at $t=1$ : $a_T = 4/\sqrt5 \approx 1.79$ and $a_N = 2/\sqrt5 \approx 0.89$ .
In a car, $a_T$ is the push into your seatback from speeding up or braking, while $a_N$ is the sideways push toward the door as you round a curve.