## Motion Defined Parametrically With Non Constant Acceleration

Suppose a a particle is moving along a curve with Cartesian coordinates given parametrically in terms of time
$t$
by
$x=\sqrt{t^2+1}, \; y= \sqrt{2t}$
then the distance of the particle from the origin is
$r= \sqrt{x^2+y^2}= \sqrt{t^2+1+2t}=t+1$
.
The velocity is
$\begin{pmatrix} \ dx/dt\\ dy/dt\end{pmatrix} = \begin{pmatrix} \frac{t}{\sqrt{t^2+1}} \\ \frac{1}{\sqrt{2t}} \end{pmatrix}$
and the acceleration is
$\begin{pmatrix} \ d^2x/dt^2 \\ d^2y/dt^2\end{pmatrix} = \begin{pmatrix} \frac{}{(\sqrt{t^2+1})^3} \\ - \frac{1}{2 \sqrt{2t}} \end{pmatrix}$
.
The speed is
$\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2} = \sqrt{\frac{2t^3+t^2+1}{2t(t^2+1)}}$
and the magnitude of the acceleration is
$\sqrt{(\frac{d^2x}{dt^2})^2+(\frac{d^2y}{dt^2})^2} = \sqrt{\frac{8t+(t^2+1)^3}{8t(t^2+1)^3}}$
.