A rocket is fired vertically from a launch pad $B$. Its flight is tracked by radar from point $A$. When the rocket is at point $P$, the angle of elevation is $\theta = 47.0^{\circ}$. At a time of $0.5$ $s$ later, the angle is $\theta = 48.0^{\circ}$. Knowing that $b = 4$ $km$, determine approximately the speed of the rocket during the $0.5$ second interval.

A. $151$ $m/s$
B. $205$ $m/s$
C. $306$ $m/s$
D. $325$ $m/s$
E. $341$ $m/s$