Calculate linear momentum (p = mv), elastic and inelastic collisions, kinetic energy transfer, and conservation of momentum. Includes collision type reference.
Momentum is the product of mass and velocity, and in a closed system it is conserved through collisions and separations. That makes it one of the most useful quantities for describing impacts, recoils, and other motion-transfer problems.
This calculator handles both single-object momentum and two-body collision cases. For collisions, it can compare elastic and perfectly inelastic outcomes, compute the post-collision velocities, and show the kinetic-energy change alongside the momentum total.
That makes it useful when the question is not just "how much motion is there?" but "how does that motion change after the interaction?"
Momentum problems often become tedious because the algebra depends on sign convention, collision type, and whether kinetic energy is conserved. This page keeps the conservation law, the collision formulas, and the energy change together so you can compare scenarios without rebuilding the setup each time.
Momentum: p = mv Conservation: m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂' Elastic Collision: v₁' = [(m₁−m₂)v₁ + 2m₂v₂] / (m₁+m₂) v₂' = [(m₂−m₁)v₂ + 2m₁v₁] / (m₁+m₂) Perfectly Inelastic: v_f = (m₁v₁ + m₂v₂) / (m₁+m₂) Kinetic Energy: KE = ½mv²
Result: v_combined = 8.33 m/s, KE lost = 50%
A 1500 kg car at 60 km/h (16.67 m/s) rear-ending a stationary 1500 kg car in a perfectly inelastic collision results in both cars moving at 8.33 m/s (30 km/h). Half of the kinetic energy is lost to deformation and heat — demonstrating why crashes are so destructive.
Momentum conservation is one of the deepest principles in physics, arising from the translational symmetry of space (Noether's theorem). Unlike energy, which can change form, momentum is always conserved in a closed system — making it the most reliable tool for analyzing collisions and interactions.
Accident reconstruction engineers use momentum conservation to determine pre-collision speeds from post-collision evidence (skid marks, deformation, final positions). The equations are solved in both x and y directions for oblique (angled) collisions, and the coefficient of restitution is estimated from vehicle damage severity.
In particle accelerators, momentum conservation determines what products are possible in a collision and what angles they emerge at. Four-momentum (combining energy and 3D momentum into a relativistic four-vector) is the conserved quantity in special relativity, and its invariant mass is used to identify new particles.
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By Newton's third law, internal forces between colliding objects are equal and opposite, so they cancel when summing momentum. External forces (gravity, friction) are negligible during brief collisions, so total momentum is conserved.
Elastic collisions conserve both momentum and kinetic energy, so the objects bounce apart. Inelastic collisions conserve momentum but lose some kinetic energy to heat, sound, and deformation. In a perfectly inelastic collision, the objects stick together.
In explosions (internal energy release), KE increases while momentum is still conserved. In ordinary collisions, KE either stays the same (elastic) or decreases (inelastic).
The coefficient of restitution e = (v₂' − v₁') / (v₁ − v₂) measures how "bouncy" a collision is. e = 1 is perfectly elastic, e = 0 is perfectly inelastic, and real collisions fall between.
Rockets use conservation of momentum: exhaust mass goes backward with high velocity, so the rocket gains forward momentum. The rocket equation (Tsiolkovsky) extends this to continuous mass ejection.
Angular momentum (L = Iω or L = r × p) is separately conserved in the absence of external torques. This calculator handles linear momentum only.