How to Calculate Momentum: Complete Physics Guide
Learn momentum p = mv, impulse, and conservation in collisions and explosions. Worked examples cover cars, bullets, and ice skaters, with common mistakes and the Momentum Calculator.
How to Calculate Momentum
Momentum quantifies "motion content": how hard it is to stop an object. A slow truck and a fast baseball can carry similar momentum; the truck's mass compensates for lower speed. Newton's laws in their modern form are statements about how forces change momentum.
Momentum is a vector. Direction matters as much as magnitude — head-on collision analysis fails if you treat momenta as unsigned numbers.
The core formula
\vecp = m\vecv
Magnitude:
p = mv
Where m is mass in kilograms, v is speed in m/s, and p is in kg·m/s (equivalently N·s).
Doubling mass doubles momentum; doubling velocity doubles momentum. Unlike kinetic energy (\propto v^2), momentum scales linearly with speed.
Impulse and Newton's second law
Force is the rate of change of momentum:
\vecF_\mathrmnet = \fracd\vecpdt
For constant mass, this becomes \vecF = m\veca. Integrated over a time interval, the impulse equals the change in momentum:
\vecJ = ∫ \vecF\,dt = Δ\vecp = \vecp_f - \vecp_i
For an average force lasting time Δ t:
\vecF_\mathrmavgΔ t = Δ\vecp
Airbags and crumple zones increase Δ t so that the same Δ p requires a smaller average force — safer for occupants.
Conservation of momentum
If the net external force on a system is zero (or negligible during a short collision), total momentum is conserved:
Σ \vecp_\mathrmbefore = Σ \vecp_\mathrmafter
Internal forces (the push between colliding cars) cancel in pairs by Newton's third law and do not change the system's total momentum.
Elastic vs inelastic collisions
"Conserved momentum" does not mean "conserved kinetic energy." Always check which laws apply.
Worked example: single-object momentum
Problem. A 1,500 kg car travels at 20 m/s east. Find its momentum.
p = (1500)(20) = 30,000 \text kg·m/s east
Same mass at 20 m/s west has momentum of equal magnitude in the opposite direction — they cancel if those two cars meet in a symmetric head-on case.
Worked example: perfectly inelastic collision
Problem. Car A (1,000 kg) at 15 m/s hits Car B (1,200 kg) at rest; they stick. Find the final speed (1D).
m_A v_A + m_B (0) = (m_A + m_B)v_f
v_f = (1000 × 15)/(2200) = 6.82 \text m/s
in the original direction of A.
Kinetic energy before: (1)/(2)(1000)(15)^2 = 112,500\,\mathrmJ.
After: (1)/(2)(2200)(6.82)^2 ≈ 51,200\,\mathrmJ.
Roughly half the KE was lost to deformation and heat — typical of a sticky collision — while momentum stayed conserved.
Worked example: recoil
Problem. A 0.020 kg bullet leaves a 3.0 kg rifle at 400 m/s. Find the rifle's recoil speed if it was initially at rest (ignore the shooter's hold).
0 = m_b v_b + m_r v_r
v_r = -(m_b v_b)/(m_r) = -(0.020 × 400)/(3.0) = -2.67 \text m/s
The minus sign means opposite the bullet. Real rifles transfer recoil into the shooter's body; the isolated-gun ideal still teaches the momentum balance.
Worked example: 1D elastic collision (equal mass)
When two equal masses collide elastically in one dimension and the target is at rest, they exchange velocities: the incoming mass stops; the target takes v_i. More generally, for masses m_1, m_2 with u_2 = 0:
v_1 = u_1(m_1 - m_2)/(m_1 + m_2), \quad v_2 = u_1(2m_1)/(m_1 + m_2)
Example. m_1 = 2\,\mathrmkg, m_2 = 1\,\mathrmkg, u_1 = 3\,\mathrmm/s:
v_1 = 3×(1)/(3) = 1\text m/s, \quad v_2 = 3×(4)/(3) = 4\text m/s
Impulse example: hitting a ball
Problem. A 0.15 kg baseball at -30\,\mathrmm/s (toward the bat) leaves at +40\,\mathrmm/s. Contact lasts 0.002 s. Find average force on the ball.
Δ p = m(v_f - v_i) = 0.15(40 - (-30)) = 0.15 × 70 = 10.5 \text kg·m/s
F_\mathrmavg = (Δ p)/(Δ t) = (10.5)/(0.002) = 5,250 \text N
Large forces over short times are the essence of hitting and kicking.
Two dimensions
Conserve momentum in each component separately:
Σ p_x,\mathrmi = Σ p_x,\mathrmf, \quad Σ p_y,\mathrmi = Σ p_y,\mathrmf
Billiard-ball and traffic-collision reconstructions use this component split when angles are measured from skid marks or video.
Centre of mass frame
Total momentum equals total mass times centre-of-mass velocity:
\vecP_\mathrmtotal = M\vecv_\mathrmcm
If \vecP_\mathrmtotal is constant, \vecv_\mathrmcm is constant. Collisions look simpler in the CM frame: momenta are equal and opposite before and after for two-body isolated systems.
Applications
Vehicle safety. Crash impulse and Δ t design; momentum sets how much impulse must be absorbed.
Rocket propulsion. Exhaust carries backward momentum; the rocket gains forward momentum (variable-mass form of Newton's law).
Sports. Follow-through increases contact time and can increase impulse delivered to a ball.
Astrophysics. Gravitational slingshots and decay of binary systems respect momentum conservation in the wider system.
Ballistics and forensics. Recoil, penetration, and trajectory estimates start from p = mv and impulse.
Common mistakes
Cancelling masses incorrectly. In sticky collisions, final mass is the sum; do not reuse a single mass with an averaged velocity without the right algebra.
Conserving KE automatically. Only elastic collisions conserve KE; momentum conservation is the safer starting point when external forces are negligible.
Ignoring direction / signs in 1D. Assign a positive axis and stick to it.
Using weight instead of mass. Momentum needs mass in kg, not newtons.
Applying conservation when external forces dominate. A car braking on pavement has large external friction; its momentum is not conserved as an isolated quantity — the Earth–car system is.
Confusing p with KE. Stopping distance from energy considerations is not the same problem as impulse from momentum — related, but different equations.
Frequently asked questions
Is momentum always conserved? Total momentum of an isolated system is. Open systems exchange momentum with the surroundings (thrust, friction, gravity during long times).
Why do airbags help if Δ p is fixed? Same change in momentum spread over larger Δ t means smaller average force, reducing injury risk.
Can a light object have more momentum than a heavy one? Yes — if it is fast enough. A rifle bullet can out-momentum a slowly drifting bowling ball.
What is the difference between momentum and inertia? Inertia (mass) resists changes in motion. Momentum is mass already in motion (mv). Related ideas; not identical.
How does bouncing change impulse? A ball that rebounds has a larger Δ p than one that stops and drops, because velocity reverses. The floor (or bat) must supply a larger impulse.
Do explosions conserve momentum? Yes for the fragments as a system if external forces are negligible during the blast. Chemical energy increases KE; total momentum still matches the pre-explosion momentum (often zero in the CM frame).
Summary
Momentum is:
\vecp = m\vecv
Impulse equals Δ\vecp. When external forces are negligible, total momentum is conserved — the key to collision and recoil problems. Treat momentum as a vector, do not assume kinetic energy is conserved unless the collision is elastic, and use components in two dimensions.
Calculate momentum with our Momentum Calculator.