Momentum Calculator
Momentum (kg⋅m/s)
1000
How it works
Momentum (p) is the product of an object's mass and velocity: p = m × v. It is a vector quantity — direction matters. The law of conservation of momentum states that in a closed system, total momentum before an event equals total momentum after.
**Impulse-momentum theorem** Impulse (J) = F × Δt = Δp (change in momentum). Applying a force over time changes momentum. This connects force analysis to kinematics: a 1000 N force applied for 0.01 seconds changes momentum by 10 N·s. Vehicle crash analysis uses this relationship to estimate impact forces from measured deceleration data.
**Elastic vs. inelastic collisions** Elastic collision: both momentum and kinetic energy are conserved. Billiard balls approximate elastic collisions. Inelastic collision: momentum is conserved, but kinetic energy is not — some is converted to heat, sound, and deformation. Perfectly inelastic: objects stick together after collision, maximum KE is lost. No real collision is perfectly elastic; the coefficient of restitution (0 to 1) quantifies energy retention.
**Center of mass frame** Analyzing collisions in the center-of-mass (CM) frame simplifies calculations. In the CM frame, total momentum is zero. After collision, objects move in opposite directions from CM. Transform between lab frame and CM frame for complex collision problems.
**Rockets and variable-mass systems** Rockets expel mass at high velocity to generate thrust — momentum of exhaust equals momentum gained by rocket. The Tsiolkovsky rocket equation: Δv = v_e × ln(m₀/m_f), where v_e is exhaust velocity, m₀ is initial mass, m_f is final mass. This determines how much propellant is needed for a given velocity change.
Frequently Asked Questions
- Both are conserved in elastic collisions; only momentum is conserved in inelastic collisions. Momentum (p = mv) is a vector — direction matters. KE (½mv²) is scalar. In a perfectly inelastic collision (two objects stick together), momentum is conserved but KE loss can be calculated: the 'lost' KE becomes heat, sound, and deformation. Example: 1000 kg car at 20 m/s hits a stationary 1000 kg car, they stick: p_final = 1000×20 = 20,000 N·s, v_final = 10 m/s. KE_before = 200,000 J, KE_after = 100,000 J — half the kinetic energy is 'lost.'
- A rocket expels exhaust mass at high velocity, gaining momentum in the opposite direction. In free space with no external forces, total momentum is conserved. As mass is expelled (dm at velocity -v_e relative to rocket), the rocket gains d(mv) = v_e × dm momentum. Integrating: Δv = v_e × ln(m₀/m_f) — the Tsiolkovsky equation. To reach low Earth orbit (Δv ≈ 9.5 km/s) with exhaust velocity of 4.5 km/s (hydrogen/oxygen engine): mass ratio m₀/m_f = e^(9.5/4.5) = 8.3. Over 87% of launch mass must be propellant — why orbital rockets look mostly like fuel tanks.
- Angular momentum L = I × ω (moment of inertia × angular velocity). Conservation: if no external torque acts, L is constant. A spinning figure skater pulling in their arms reduces I → ω must increase to keep L constant → they spin faster. Gyroscopes resist changes in orientation because angular momentum has a fixed direction in space. Earth's axial precession (25,772-year cycle) is caused by the Moon and Sun's gravitational torques slowly changing Earth's angular momentum vector direction. Conservation of angular momentum is as fundamental as linear momentum conservation.
- Impulse = F × Δt = change in momentum. A vehicle going from 60 mph to 0 mph: Δp = 1400 kg × 26.8 m/s = 37,500 N·s. If stopped in 0.1 s (rigid wall): F = 37,500/0.1 = 375,000 N = 84,000 lbf. If stopped in 0.4 s (crumple zone): F = 37,500/0.4 = 93,750 N — 4× less force. This is why crumple zones, airbags, and deformable barriers all reduce peak force by extending the stopping time — the impulse (momentum change) is fixed by initial speed, but force is reduced by increasing Δt.