No question. I'll assume you want the instantaneous speed of the projectile launcher when firing the projectile.
The projectile has a momentum equal to the product of its mass and speed.
So, p = 0.08 • 300
p = 24 kgm/s where p is the momentum of the projectile
The initial total momentum of the system was zero, so the projectile launcher must also have a momentum of 24 kgm/s after firing in the opposite direction to the projectile, due to Newton's Third Law of motion.
24 = 3 • v
v = 24/3
v = 8 m/s
Therefore the projectile launcher has a recoil speed of 8 m/s after firing the projectile.