Answer
(a) The speed of the laser beam is $~~c~~$ according to the Galaxa's measurement.
(b) The speed of the missile relative to the Galaxa is $0.80~c$
Work Step by Step
(a) Since the laser beam is an electromagnetic wave, the speed in a vacuum measured by all observers in all reference frames will be $c$. The speed of the laser beam is $c$ according to the Galaxa's measurement.
(b) Let $v_{MG}$ be the velocity of the Millenia relative to the Galaxa. Then $v_{MG} = 0.55~c$
Let $v_{XM}$ be the velocity of the missile relative to the Millenia. Then $v_{XM} = 0.45~c$.
We can find $v_{XG}$:
$v_{XG} = \frac{V_{XM}~+~v_{MG}}{1+\frac{(v_{XM})~(v_{MG})}{c^2}}$
$v_{XG} = \frac{(0.45~c)~+~(0.55~c)}{1+\frac{(0.45~c)~(0.55~c)}{c^2}}$
$v_{BA} = \frac{1.0~c}{1.2475}$
$v_{BA} = 0.80~c$
The velocity of the missile relative to the Galaxa is $0.80~c$
Therefore, the speed of the missile relative to the Galaxa is $0.80~c$.
