Suppose you have a disc $C$ of radius $V$ with center $c$ and you randomly place a point $p$ in it. $p$ Behaves as follows: at every time-step, $p$ calculates its angle to $c$, and moves a distance of size $d_0$ at an angle of $-\pi/2 < a_0 < \pi/2$ from $c$ ($d_0$ and $a_0$ are constants for a given $p$). Having done this, it stops and draws a disc of radius $V$ around itself.
My question is this: do $a_0,d_0$ exist such that for any given $p$ inside the disc, the union of all discs $p$ draws around itself will cover the disc $C$ in its entirety after a finite number of steps?
The motion of $p$ will look a little like that of a clumsy radar around $c$, hence the question's title
Intuition: Notice that it is possible to always keep $p$ inside $C$ by setting, for example, $a_0=\pi/4$, $d_0=V/\sqrt{8}$. To see this, draw a circle $C'$ of radius $V/2$ between $p$ and $c$ (at the angle of $p$ to $c$) where $p$ is at the perimeter of the circle. Notice that moving a distance of $V/\sqrt{8}$ at $\pi/4$ degrees will move $p$ precisely to a point in the perimeter of $C'$, so at every step $p$ will have at most distance $2*(V/2)=V$ from $c$.