Momentum Calculator
Calculate linear momentum (p = mv) and impulse. Enter mass and velocity — get momentum and impulse results for collision analysis and impact engineering.
kg
m/s
Enter your values above to see the results.
Tips & Notes
- ✓Momentum is always conserved in isolated systems — no external forces. Total momentum before collision = total momentum after, regardless of whether the collision is elastic or inelastic.
- ✓Impulse = change in momentum = F × Δt. Airbags extend the collision time, reducing peak force: same impulse (Δp) over longer time means smaller F. This is why padding saves lives.
- ✓Momentum is a vector — direction matters. Head-on collision between two equal masses at equal speeds: total momentum = 0. They stop (perfectly inelastic) or reverse (elastic).
- ✓The SI unit is kg·m/s. A 70 kg person running at 5 m/s has momentum 350 kg·m/s — the same as a 0.007 kg bullet at 50,000 m/s (not realistic, but illustrates the mass-velocity tradeoff).
- ✓Angular momentum (L = Iω) is a separate but analogous concept for rotating bodies. A figure skater spins faster by pulling in arms because angular momentum L = Iω is conserved as I decreases.
Common Mistakes
- ✗Confusing momentum (p = mv) with kinetic energy (KE = ½mv²) — momentum is proportional to v, KE to v². Both are conserved in different ways: momentum always, KE only in elastic collisions.
- ✗Adding momenta as scalars when they are in different directions — two 1,000 kg cars each at 20 m/s in opposite directions have momenta +20,000 and −20,000 kg·m/s, totaling 0, not 40,000 kg·m/s.
- ✗Forgetting that impulse includes both force and time — stopping a 70 kg runner at 5 m/s requires an impulse of 350 N·s regardless of time. A wall provides this in 0.01 s (35,000 N peak force); a foam pad in 0.2 s (1,750 N — 20× gentler).
- ✗Applying conservation of momentum to systems with external forces — if friction, gravity, or external forces act during the event, total momentum changes. Conservation only applies to isolated systems.
- ✗Using mass in wrong units — momentum in kg·m/s requires mass in kg. A 2,000 lb car = 907 kg; a 5 lb object = 2.27 kg. Always convert before calculating.
Momentum Calculator Overview
Momentum is the quantity of motion — it describes how difficult it is to stop a moving object and governs all collision and impact events. Unlike kinetic energy, momentum is always conserved in isolated systems regardless of the type of collision, making it the fundamental conservation law of mechanics.
Linear momentum formula:
p = m × v | Units: kg·m/s = N·s
EX: 2,000 kg truck at 25 m/s → p = 2,000 × 25 = 50,000 kg·m/s. A 0.006 kg bullet at 900 m/s → p = 0.006 × 900 = 5.4 kg·m/s. The truck has 9,259× more momentum despite the bullet's extreme speed.Impulse-momentum theorem:
Impulse J = F × Δt = Δp = m × Δv | Units: N·s = kg·m/s
EX: Stop 1,500 kg car from 20 m/s. Δp = 1,500 × 20 = 30,000 N·s. Braking in 3s: F_avg = 10,000 N. Crash in 0.05s: F_avg = 600,000 N (60× higher force — why crumple zones matter)Conservation of momentum — collision types:
| Collision Type | Momentum | Kinetic Energy | Example |
|---|---|---|---|
| Elastic | Conserved | Conserved | Billiard balls, superball |
| Inelastic (partial) | Conserved | Partially lost | Most real collisions |
| Perfectly inelastic | Conserved | Maximum loss | Objects stick together |
| Explosion | Conserved (=0 if from rest) | Gained from chemical energy | Cannon, rocket, grenade |
| Scenario | Δp (kg·m/s) | Contact Time | Avg Force |
|---|---|---|---|
| Baseball bat hit (0.145 kg, Δv=50 m/s) | 7.25 N·s | 0.001 s | 7,250 N |
| Car airbag deployment | ~15,000 N·s | 0.05 s | 300,000 N |
| Car crumple zone (50 km/h crash) | ~20,000 N·s | 0.1 s | 200,000 N |
| Person falling 1m onto mat | ~300 N·s | 0.2 s | 1,500 N |
| Person falling 1m onto concrete | ~300 N·s | 0.01 s | 30,000 N |
Frequently Asked Questions
Momentum p = m × v, where m is mass in kilograms and v is velocity in m/s. The result is in kg·m/s (Newton-seconds). Example: a 1,500 kg car at 20 m/s → p = 1,500 × 20 = 30,000 kg·m/s. A 0.145 kg baseball pitched at 44 m/s (100 mph) → p = 0.145 × 44 = 6.38 kg·m/s. To stop either object requires an impulse equal in magnitude to its momentum.
Impulse = F × Δt = Δp (change in momentum). The impulse-momentum theorem states that the impulse applied to an object equals its change in momentum. Example: stopping a 1,500 kg car from 20 m/s (momentum = 30,000 kg·m/s): if braking takes 3 seconds, average braking force = 30,000 / 3 = 10,000 N. If an airbag extends crash time from 0.01 s to 0.1 s while the same Δp = 30,000 kg·m/s occurs, peak force decreases by a factor of 10. This is why padding and crumple zones save lives — same impulse, longer time, lower peak force.
In an isolated system (no external forces), total momentum before = total momentum after. For two objects: m₁v₁ + m₂v₂ = m₁v₁′ + m₂v₂′ (primes = after collision). Elastic collision (KE also conserved): objects bounce with no energy loss. Perfectly inelastic collision: objects stick together, maximum KE is lost. Example: 2 kg ball at 5 m/s hits stationary 3 kg ball. If they stick: (2×5 + 3×0) = 5 × v_final → v_final = 10/5 = 2 m/s. KE before = ½×2×25 = 25 J; KE after = ½×5×4 = 10 J. 15 J converted to heat, sound, deformation.
In any collision, momentum is conserved. Kinetic energy is only conserved in elastic collisions. A superball bouncing off a wall approximates elastic (KE preserved, momentum reversed for the ball). A lump of clay hitting a wall is perfectly inelastic (momentum transferred to wall/Earth system, all KE converts to deformation/heat). For two objects: elastic means you can solve for final velocities using both momentum and energy conservation equations simultaneously. Inelastic means only momentum conservation applies — you need additional information (final velocity of combined mass) to solve.
Rocket propulsion is direct momentum conservation. The rocket expels mass (exhaust gas) at high velocity backward; the rocket gains equal and opposite momentum forward. Tsiolkovsky rocket equation: Δv = v_e × ln(m_i / m_f), where v_e is exhaust velocity, m_i is initial mass, m_f is final mass. Example: rocket with exhaust velocity 3,000 m/s, propellant = 90% of initial mass → Δv = 3,000 × ln(10) = 3,000 × 2.303 = 6,909 m/s. This is why rockets must carry enormous fuel mass — the exhaust momentum must equal the final vehicle momentum.
Impact force = Δp / Δt = (m × Δv) / Δt. Example: a 75 kg person falls from 2 m height, hits the ground at 6.26 m/s (v = √(2gh) = √(2×9.81×2)). If landing on concrete (Δt = 0.01 s): Force = 75 × 6.26 / 0.01 = 46,950 N ≈ 64× body weight. If landing on a cushion (Δt = 0.2 s): Force = 75 × 6.26 / 0.2 = 2,348 N ≈ 3.2× body weight. Same momentum change, 20× longer time, 20× lower peak force — this is the engineering principle behind sports mats, vehicle crumple zones, and helmet padding.