In a collision, forces are violent, brief, and nearly impossible to track — but momentum keeps the books anyway. Whatever the objects do to each other, their total momentum after equals their total momentum before, provided no outside impulse interferes. This conservation of momentum calculator solves both textbook extremes: perfectly inelastic collisions, where the objects stick and move as one, and perfectly elastic collisions, where they bounce with kinetic energy intact — and it audits momentum and energy before and after so you can watch the conservation happen.
The Conservation Law
Momentum is mass times velocity, p = mv, a vector. In any collision inside a system closed to outside impulses, the total momentum vector is the same before and after: m₁v₁ + m₂v₂ = m₁v₁′ + m₂v₂′. This is not an approximation tuned to gentle bumps — it follows from Newton’s third law (the forces between the objects are equal, opposite, and simultaneous, so their impulses cancel) and it holds from billiard balls to galaxies merging. What varies between collisions is kinetic energy, and that variation is how collisions are classified.
Perfectly Inelastic: The Sticking Collision
If the objects collide and move as one, momentum alone fixes the outcome: vf = (m₁v₁ + m₂v₂) ÷ (m₁ + m₂). Example: a 2 kg cart at 3 m/s strikes a stationary 1 kg cart and they latch. Total momentum is 6 kg·m/s spread over 3 kg, so vf = 2 m/s. Kinetic energy drops from 9 J to 6 J — 3 J converted to deformation, heat, and sound. Inelastic collisions always lose kinetic energy; the sticking case loses the maximum that momentum conservation permits, which is why car crumple zones are engineered to approach it.
Perfectly Elastic: The Perfect Bounce
When kinetic energy is also conserved, two equations pin down both final velocities: v₁′ = ((m₁ − m₂)v₁ + 2m₂v₂) ÷ (m₁ + m₂) and the symmetric partner for v₂′. Same masses as above, elastic this time: v₁′ = (1 × 3 + 0) ÷ 3 = 1 m/s and v₂′ = (2 × 2 × 3) ÷ 3 = 4 m/s. Audit it: momentum 2 × 1 + 1 × 4 = 6 ✓, kinetic energy 1 + 8 = 9 J ✓. Equal masses in an elastic collision simply swap velocities — the Newton’s cradle effect. True elastic collisions exist between atoms and near-perfectly between billiard balls; everything else falls between the two modes of this calculator.
Real Collisions Live Between the Extremes
The coefficient of restitution (e = relative speed after ÷ relative speed before) locates a real collision on the spectrum: e = 1 is elastic, e = 0 is perfectly inelastic, and a car crash manages e ≈ 0.1 to 0.3 while a basketball bounce reaches about 0.8. If your problem gives a restitution coefficient or says the objects “collide and separate with these energies,” neither calculator mode applies directly — solve momentum plus the restitution relation together. The two modes here are the bookends every real answer must fall between, which makes them a powerful sanity check.
Explosions: Conservation Running Backwards
Run the film in reverse and a sticking collision is a single object at rest splitting apart — a firework, a recoil, a decaying nucleus. Total momentum starts at whatever the parent had (zero, if stationary) and the fragments’ momenta must sum to it: a 3 kg shell bursting into 2 kg and 1 kg pieces sends them oppositely with momenta in the inverse mass ratio, the lighter piece moving twice as fast. Recoil calculations for firearms and rocket staging are exactly this bookkeeping.
Common Mistakes
Forgetting velocities are vectors is the big one: head-on motion means opposite signs, and a total momentum of 6 − 4 = 2, not 10. Expecting kinetic energy to survive a sticking collision is the second — momentum is the conserved quantity there, never energy. Third: applying conservation across a collision with a wall or the ground without including the Earth in the system — the wall delivers an external impulse, and the visible momentum is not conserved (the Earth takes the difference, immeasurably slowly). Keep the system closed, the signs consistent, and the audit the calculator prints will agree with your algebra.
Related Tools
The Momentum Calculator covers single-object p = mv, and the Physics Calculators hub holds the full mechanics collection including the Newton’s Second Law Calculator.
Frequently Asked Questions
How do you calculate final velocity in an inelastic collision?
Divide total momentum by total mass: vf = (m1v1 + m2v2) / (m1 + m2). A 2 kg cart at 3 m/s sticking to a stationary 1 kg cart gives 6 / 3 = 2 m/s.
Is kinetic energy conserved in collisions?
Only in perfectly elastic collisions. Inelastic collisions convert some kinetic energy to deformation, heat and sound; the sticking case loses the most momentum conservation allows. Momentum itself is conserved in all of them.
What happens when equal masses collide elastically?
They exchange velocities. A moving mass stops dead and hands its full velocity to the stationary one — the behavior on display in a Newton’s cradle.
Why is momentum always conserved but energy is not?
The internal forces of a collision come in equal-opposite pairs acting for the same time, so their impulses cancel exactly — momentum has nowhere to go. Kinetic energy, a scalar tied to deformation, can convert into heat and damage; total energy is conserved, but the kinetic share is not.
How do signs work in momentum problems?
Pick a positive direction; velocities against it are negative. A 2 kg object at +3 m/s meeting a 1 kg object at −4 m/s has total momentum 6 − 4 = 2 kg m/s, not 10.
What is the coefficient of restitution?
The ratio of separation speed to approach speed along the collision line. One is perfectly elastic, zero is perfectly inelastic, and real collisions — car crashes around 0.1 to 0.3 — live between the two modes this calculator solves.
Does conservation of momentum apply to explosions?
Yes — it is the sticking collision in reverse. The fragments’ total momentum equals the original object’s, which is how recoil and rocket staging are calculated.