The principle of the automatic weapon is based on the utilization of the phenomenon of recoil which takes place on firing: the breech-block moves backwards after firing and compresses the spring, which in turn brings the reloading mechanism into action. Find what must be the velocity of the bullet for the breech-block to move back a distance a, if the mass of the bullet be m, the mass of the breech-block to move back a distance a, if the mass of the bullet be m, the mass of breech-block M and the coefficient of elasticity of the spring k. Neglect the mass of the charge.
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Sol. For the breech-block to move back a distance a, it is necessary that the work done in overcoming the elastic force of the spring should be ka 2 /2. This work will be done at the expense of the kinetic energy which the breech-block acquires from the recoil. If the breech-block has initial velocity u, its kinetic energy Mu 2 /2 = ka 2 /2, hence u = a 
. On the other hand, the absolute momentum Mu of the breech-block on firing must equal the momentum of the bullet, mv (since they are opposite in direction and must give a sum of zero). Therefore
v = 
= 
.
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