Among other things, the angular speed of a rotating vortex $($such as in a tornado$)$ may be determined by the use of Doppler weather radar. A Doppler weather radar station is broadcasting pulses of radio waves at a frequency of $2\mathrm{.}85$ GHz$,$ and it is raining northeast of the station. The station receives a pulse reflected off raindrops, with the following properties: the return pulse comes at a bearing of $51\mathrm{.}4$ north of east; it returns $180$s after it is emitted; and its frequency is shifted upward by $296$ Hz$.$ The station also receives a pulse reflected off raindrops at a bearing of $52\mathrm{.}3$ north of east, after the same time delay, and with a frequency shifted downward by $296$ Hz$.$ These reflected pulses have the highest and lowest frequencies the station receives.

$($a$)$

Determine the radial$-$velocity component of the raindrops $($in m$/$s$)$ for each bearing $($take the outward direction to be positive$).$

$51\mathrm{.}4$ north of east m$/$s$52\mathrm{.}3$ north of east m$/$s

$($b$)$

Assuming the raindrops are swirling in a uniformly rotating vortex, determine the angular speed of their rotation $($in rad$/$s$).$