The derivation is performed below...
The desired result for the velocity is
In applying conservation of energy we will start from the configuration in which the arrow is drawn back and stationary in the bow.
Its potential energy (PE) at this point is
Even though it gains considerable speed upon release, before finally slowing again to a halt we can "skip over" all of this because the problem does not concern itself with this part of the motion. This is one of the beauties of using conservation of energy in solving problems in mechanics.
When it comes to rest at height h, its energy is all gravitational PE, given by
Cons of energy insists that
Solve for h:
For the second result, we note that the arrow drops only a distance h in returning to "head level" (as opposed to h+d). The PE decrease is
Cancel m, and solve for v: