Conservation of momentum states that the vector sum of momentum of a system does not change when the net external impulse is negligible. It is the workhorse principle for collisions, explosions, recoil, and short-time interactions.
Core statement
For a system of particles:
P = Σ p_i = Σ (m_i v_i)
P_before = P_after
Condition: external impulse
The exact relation is:
ΔP = J_ext = ∫ F_ext dt
If the interaction time is short enough that J_ext ≈ 0, total momentum is conserved during that interval. This is why we often treat collisions as momentum-conserving even though gravity and friction exist.
Why internal forces cancel
Internal forces come in action–reaction pairs. When summing over all particles, internal forces cancel, and only external forces remain in dP/dt.
1D two-body collision (with restitution)
The coefficient of restitution e is defined by the relative speed along the line of impact:
e = (relative speed after)/(relative speed before)
= -(v2 - v1)/(u2 - u1)
Together with momentum conservation, the post-collision velocities are:
v1 = (m1 - e m2)/(m1 + m2) * u1 + (1 + e) m2/(m1 + m2) * u2
v2 = (1 + e) m1/(m1 + m2) * u1 + (m2 - e m1)/(m1 + m2) * u2
Elastic: e = 1. Perfectly inelastic (stick together): e = 0.
Center-of-mass view
Total momentum is directly related to center-of-mass motion:
V_cm = (Σ m_i v_i) / M
P = M V_cm
If external impulse is negligible, V_cm is constant. Collisions then look like internal rearrangements around a uniformly moving center of mass.
Common pitfalls
- Momentum conservation does not mean each object's momentum stays the same.
- Momentum is a vector (a signed quantity in 1D), so large momenta can cancel.
- Kinetic energy can change even when momentum is conserved (inelastic collisions).