Alvin, you over simplified your equations. Initially the bolt and the Barrel/reciever are locked together, and recoil at the same velocity. The equation gets complicated when the toggle knobs hit the kickup on the frame, but the higher kinetic energy of the longer barrel assembly will be forcing the toggle block open, NOT just the weight of the toggle itself. What you need to focus on is the velocity term, which you dropped out. The difference between a carbine barrel recoil and a standard barrel recoil, if there is any, is in the possibly slower recoil of the the loger heavier barrel assembly. The effect of the weight on lower acceleration of the recoiling mass against the effect of the longer barrel on increasing the bullet velocity.
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