Explosions and Recoil

Explosions and recoil events involve internal forces that push system components apart. According to Newton's Third Law, these internal forces occur in action-reaction pairs that sum to zero, leaving the total linear momentum of the system unchanged. Provided that external forces are negligible compared to interaction forces, the system is approximately isolated during the event and momentum is conserved.

Kinetic energy, however, is not conserved in explosions and recoil. Instead, internal energy from explosives or springs are converted to kinetic energy.

Consider a cannon of mass $m_c$ containing a projectile of mass $m_p$. Initially, both components are at rest, so the total initial momentum $\vec{P}_i$ is zero:

Upon firing, the projectile launches with velocity $\vec{v}_p$ and the cannon recoils with velocity $\vec{v}_c$. The final total momentum $\vec{P}_f$ is the vector sum of the individual momenta:

Applying the conservation of momentum $\vec{P}_f = \vec{P}_i$ yields:

Rearranging this equation relates the recoil momentum to the projectile momentum:

This result demonstrates that the momentum of the cannon is equal in magnitude and opposite in direction to the momentum of the projectile. Since the cannon mass $m_c$ is typically much larger than the projectile mass $m_p$, the recoil speed $|\vec{v}_c|$ is significantly smaller than the projectile speed $|\vec{v}_p|$.

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