It's all about kinetic energy KE = (1/2)* m * V^2 . Any textbook discussion on "dynamics" or "mechanics" should have those equations.
You could calculate from what height your 7.86 lbf weight must fall under the influence of gravity to generate the correct velocity to produce the KE (Joules). Then use an air cylinder to raise it back up. You'd have to size the cylinder properly. Just make sure your UNITS are correct.
Can't remember the motion equations. S = (1/2)*a*t^2 is one, but there's another that can be solved for velocity. At least my fuzzy brain thinks there is.
If your rig can't
DROP the weight (or, equivalently, use a pendulum) then you must do calculating to determine:
(1) how fast you must stroke the air cylinder to get the velocity required
(2) how big (diameter) the cylinder must be at a specified air pressure to get enough force on the piston in order to achieve the acceleration and velocity required.
THEN, after all that cypherin', you'll probably just have to test it and experimentally measure the results anyway. Good luck
TygerDawg