so assuming constant force while accellerating:
So assume a 1m pull on the bow, we have the time on the bow is:
and so the speed at which the arrow leaves the bow is:
Hence doubling the force increases the speed by a factor of sqrt(2).
Alternativly, work done = change in KE = force*distance.
Hence doubling the force doubles the KE, but KE is proportional to the
square of speed, so speed incread=ses by a factor of sqrt(2).