Question $2$: Motion in two dimensions

Let the position vector $r_t$ of a point moving along a path in the Cartesian plane be given by

$r_t=x(t)i_{+}y(t)j_{?}=t^4{i}_{+}18t^2{j}_{.}$

$($a$)$ Calculate the velocity $v_t$ and the acceleration $a_t.$

$($b$)$ Let $x(t)$ and $y(t)$ be given in units of metres and $t$ in s$.$ Show that the magnitude of $a(t)$ is at least $36m{s}^{-2}$ at all times and that $a(2s)=60m{s}^{-2}.$

$($c$)$ Show that the trajectory of the point is given by the curve

$y(t{)}^2=324x(t).$

What type of curve is this?

$($d$)$ Show that the velocity vector has angle of $45$ to the i and $j$ axes for $t=3s.$