Momentum and collision calculator: final velocities, total momentum and kinetic energy lost in an elastic, inelastic or partly elastic head-on collision.
By AdminDrafted with AI writing toolsUpdated Checked examples: 5
In this elastic collision momentum stays at 5 kg·m/s; body 1 leaves at 0.333333 m/s and body 2 at 4.33333 m/s, and 0% of the kinetic energy is lost.
e = 1 keeps all kinetic energy; e = 0 means the bodies move off together.
In a collision between two bodies, the total momentum m₁u₁ + m₂u₂ is the same before and after. The coefficient of restitution e, the ratio of separation speed to approach speed, supplies the second equation: e = 1 is a perfectly elastic collision and e = 0 means the bodies stick together. The two equations give both final velocities, and comparing ½mv² before and after gives the kinetic energy lost.
The default, a 2 kg body at 3 m/s meeting a 1 kg body moving at −1 m/s elastically, sends them off at 0.333 m/s and 4.333 m/s. A 1,000 kg car at 20 m/s that locks onto a parked 1,500 kg car moves off at 8 m/s, and 60% of the kinetic energy goes into deformation, heat and sound.
Motion is along one line: choose a positive direction and give velocities the other way a minus sign. External forces such as road friction are taken as negligible during the impact.
Momentum is mass times velocity: p = mv, in kg·m/s. Velocity has a direction, so momentum does too: a 2 kg cart at 3 m/s has 6 kg·m/s, and a 1 kg cart moving the other way at −1 m/s has −1 kg·m/s. Together they carry 5 kg·m/s.
Choose one direction as positive and give every velocity the other way a minus sign. OpenStax's linear momentum lesson defines momentum as this vector quantity.
When no outside force acts during the impact, total momentum is the same before and after: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. That is one equation with two unknowns, the final velocities v₁ and v₂, so a second equation is needed.
The coefficient of restitution e supplies it: v₂ − v₁ = e(u₁ − u₂). The bodies separate at e times their approach speed, from e = 1 for an elastic collision down to e = 0 when they stick together.
Solving the pair gives:
Masses are in kg, velocities in m/s, total momentum in kg·m/s and kinetic energy lost in joules.
| m1 | u1 | m2 | u2 | e | v1 | v2 | Total p | KE lost |
|---|---|---|---|---|---|---|---|---|
| 2 | 3 | 1 | -1 | 1 | 0.333333 | 4.333333 | 5 | 0 |
| 1 | 2 | 1 | 0 | 1 | 0 | 2 | 2 | 0 |
| 1000 | 20 | 1500 | 0 | 0 | 8 | 8 | 20000 | 120000 |
| 2 | 3 | 1 | -1 | 0.5 | 1 | 3 | 5 | 4 |
| 2 | 5 | 2 | -5 | 0 | 0 | 0 | 0 | 50 |
An elastic collision conserves kinetic energy as well as momentum. With e = 1 the formulas become v₁ = [(m₁ − m₂)u₁ + 2m₂u₂] ÷ (m₁ + m₂) and v₂ = [2m₁u₁ + (m₂ − m₁)u₂] ÷ (m₁ + m₂).
Equal masses swap velocities, as in row 2 and in a Newton's cradle. A light body bounces back off a heavy one: 1 kg at 2 m/s striking 10 kg at rest rebounds at −18/11 ≈ −1.64 m/s, and the heavy body moves off at 4/11 ≈ 0.36 m/s.
OpenStax's collision types lesson compares elastic and inelastic cases.
The bodies stick and share one velocity, v = (m₁u₁ + m₂u₂) ÷ (m₁ + m₂). Momentum is conserved, but kinetic energy is not: the loss becomes deformation, heat and sound.
For the cars in row 3, kinetic energy falls from ½ × 1,000 × 20² = 200,000 J to ½ × 2,500 × 8² = 80,000 J. The 120,000 J difference is 60% of the original energy.
Most real collisions fall between the two extremes. Row 4 repeats row 1 with e = 0.5: the bodies leave at 1 m/s and 3 m/s, still carrying 5 kg·m/s, and 4 J of the 9.5 J of kinetic energy is lost.
For a ball dropped on a rigid floor, e = √(bounce height ÷ drop height). A ball dropped from 1 m that bounces to 0.64 m has e = √0.64 = 0.8.
Impulse is a body's change in momentum, J = m(v − u), in N·s. In row 1, body 2 goes from −1 m/s to 13/3 m/s, so it receives 1 × (13/3 + 1) = 16/3 ≈ 5.33 N·s; body 1 receives the same impulse in the opposite direction.
Average force is impulse divided by contact time. If that impact lasted 0.01 s, the average force on each body would be about 533 N. Spreading the same impulse over a longer time, as crumple zones and airbags do, lowers the force.
For kinetic energy on its own, use the work, energy and power calculator. The kinematics calculator handles motion before and after the impact, and the force, friction and incline calculator handles the friction that slows the bodies afterward.
Checked against: OpenStax UP1 §9.4: equal-mass elastic collision exchanges velocities
Checked against: v = 20000/2500 = 8; KE 200000 → 80000 J
Checked against: v1 = (5 + 1·0.5·(−4))/3 = 1, v2 = (5 + 2·0.5·4)/3 = 3; KE 9.5 → 5.5 J
Checked against: v1 = 9/11, v2 = 20/11
Both conserve momentum; only an elastic collision also conserves kinetic energy. In a perfectly inelastic collision the bodies stick together and lose the most kinetic energy that momentum conservation allows. When a 1,000 kg car at 20 m/s locks onto a parked 1,500 kg car, momentum stays at 20,000 kg·m/s while kinetic energy falls from 200 kJ to 80 kJ. Most real collisions fall between the two extremes.
It is the relative speed after a collision divided by the relative speed before, e = (v₂ − v₁) ÷ (u₁ − u₂), a number from 0 to 1. For a ball dropped onto a rigid floor, e = √(bounce height ÷ drop height). The ITF requires a type 2 tennis ball dropped from 254 cm onto concrete to rebound 135–147 cm, which corresponds to e between 0.73 and 0.76.
Momentum is mass times velocity, p = mv, measured in kg·m/s. It has a direction, so velocities in opposite directions carry opposite signs. A 1,000 kg car at 20 m/s has 20,000 kg·m/s. With no outside forces the total is the same before and after a collision, which is why two 2 kg carts meeting head-on at 5 m/s and sticking together stop dead.
They swap velocities. A 1 kg ball at 2 m/s hitting an identical ball at rest stops, and the second ball leaves at 2 m/s with all the kinetic energy; a Newton's cradle shows the same effect. With unequal masses the lighter body leaves faster: a 10 kg ball at 1 m/s sends a 1 kg ball off at 1.82 m/s and slows to 0.82 m/s.
This calculator includes 5 worked examples with independently sourced answers. They run in the test suite, and you can run them here too.