Momentum conservation

Momentum conservation states that total momentum stays constant when the net force exerted from outside the chosen collection is zero. Use a closed collection, meaning the same fixed-mass point masses are followed throughout, and an inertial frame. ThenTo see why, sum Newton's second law over the members. If ${\mathbf{F}}_{ij}$ is the force on member $i$ from member $j$, Newton's third law assumes ${\mathbf{F}}_{ij}=-{\mathbf{F}}_{ji}$. Each pair cancels in the total:The internal forces act on different bodies and can change their individual momenta; their contributions cancel only in the collection's sum. An isolated system has no external interaction, but balanced external forces also suffice for momentum conservation. A closed system merely excluding matter exchange need not have conserved momentum. ^[ref]

The usefulness is that the detailed internal forces during a collision need not be known. Both an elastic collision and a perfectly inelastic collision obey the same isolated momentum balance; kinetic energy conservation is an additional condition for the elastic collision. If the external impulse is small but nonzero, the total momentum change is correspondingly small. For a variable-mass system exchanging matter, the entering or leaving momentum must also be included.

In a translation-invariant field theory, integrating over the spacetime zero mode enforces momentum conservation at interactions and scattering amplitudes.

If a mass m moving at u splits into equal fragments at velocities $u\pm w$, conservation of momentum fixes their centre-of-mass velocity. Their kinetic energy increases by $m{w}^2/2$, independent of u. Thus an internal release energy Q gives $w=\sqrt{2Q/m}$. A front fragment must exceed a leading target's speed to close a pre-existing separation; merely matching speed does not intercept it. This kinematic result does not specify subsequent capture or braking.

For downward pre-ejection speed $v_0$, total mass $M_0$, payload mass $m_p$, and downward payload speed $v_p$ in the original inertial frame, the remaining body's upward speed is $(m_p{v}_p-M_0{v}_0)/(M_0-m_p)$. When the numerator nearly cancels, distinguishing total initial mass from final body mass is essential.

Recoil is the motion acquired by a body when it exchanges momentum with an emitted, absorbed or scattered particle. For isolated emission from a body initially at rest, momentum conservation makes the remaining body acquire the opposite momentum. If its final mass is $M$ and the recoil is nonrelativistic, its kinetic energy is $E_R\simeq p^2/(2M)$. A photon of energy $E_{\gamma }$ has momentum magnitude $E_{\gamma }/c$, so $E_R\simeq E_{\gamma }^2/(2M{c}^2)$. The transition energy is shared between the photon and the recoiling body; nuclear recoil is an important example.