Momentum (linear momentum) is a vector quantity that captures how hard it is to stop or redirect a moving body. In Newtonian mechanics it is defined as mass times velocity. A force changes momentum through impulse, and the total momentum of an isolated system is conserved.
Definition and units
For constant mass in Newtonian mechanics:
p = m v
- p: momentum (vector)
- m: mass (scalar)
- v: velocity (vector)
The SI unit is kg·m/s, which is equivalent to newton-second:
1 kg·m/s = 1 N·s
Newton's 2nd law in momentum form
The most general Newtonian statement for a system of particles is:
P = Σ p_i
F_ext,total = dP/dt
When mass is constant for a single body, this reduces to F = m a. The momentum form is useful because it stays valid for multi-particle systems and highlights what matters for conservation: external impulse.
Impulse and collisions
Impulse J is the time-integral of force. It equals the change in momentum:
J = ∫(t1→t2) F dt = Δp
F_avg = Δp / Δt
In collisions the contact force can be very large, but the collision time is short. For a given Δp, increasing the interaction time Δt reduces the average force, which is why cushioning (airbags, padding) lowers peak force.
Conservation of momentum
If the net external impulse on a system is negligible during an interaction, total momentum is conserved:
P_before = P_after
(1D) m1 u1 + m2 u2 = m1 v1 + m2 v2
Momentum conservation holds for both elastic and inelastic collisions, while kinetic energy generally does not (inelastic cases convert part of kinetic energy into heat, sound, deformation, etc.).
Center of mass connection
Total momentum relates directly to center-of-mass motion:
P = M V_cm
V_cm = (Σ m_i v_i) / M
This separates the "bulk" translation of the system (center of mass) from internal motions.
Momentum flux (fluids and jets)
When mass flows through a control volume, force is tied to momentum flow rate. A common engineering form is:
F ≈ m_dot (v_out - v_in)
Pressure can also be interpreted as momentum flux per unit area in microscopic particle pictures.
Relativistic momentum
At speeds close to the speed of light c:
p = γ m v
γ = 1 / sqrt(1 - v^2/c^2)
E^2 = (pc)^2 + (mc^2)^2
For photons (m = 0): p = E/c, and with quantum relation E = h f this gives p = h/λ.
Generalized and quantum momentum
In Lagrangian mechanics, the canonical (generalized) momentum is:
p_i = ∂L / ∂q_dot_i
In quantum mechanics, momentum becomes an operator:
p̂ = - i ħ ∇
Δx Δp ≥ ħ/2
Momentum is also deeply connected to spatial translation symmetry (Noether's theorem).
Common confusions
- Momentum depends linearly on speed, while kinetic energy depends on speed squared.
- Momentum has direction (sign in 1D); cancelling can happen even when speeds are large.