[Illuminated One]

Well, firstly of course the material matters. That's why noone built bows from stone or ice.
Of course if you are saying when I have a bow made from material x that shoots an arrow weighting y with the speed z and another bow from material q that does exactly the same thing with the same arrow than yes, the material matters not (ignoring all other things that might be important for a soldier like weight or ruggedness), then yeah, but that's a truism.

And it is not quite as simple as (energy put in)*(effeciency) = 1/2 (mass of projectile)*(speed of projectile)².
You'll have a maximum velocity that you can archieve, as that's the maximum velocity with that the bow snaps back into shape. This is dependent on material and bow shape and weight but bugger me if I can provide a formula. (Still, it's easy enough to verify: you can't throw a table tennis ball faster than a golf ball because your arm is at full speed then already).
You'll still put in the same amount of energy into the bow but that isn't imparted on the arrow but on the bow.