A 3 kg object at 4 m/s hits a stationary 3 kg object elastically. What are the final velocities?

The correct answer is: v₁' = 0 m/s, v₂' = 4 m/s. For equal masses in elastic collision: v₁'=0, v₂'=v₁ (momentum transfers completely).

A 4 kg cart at 6 m/s collides and sticks to a 2 kg cart at rest. Find final velocity.

The correct answer is: 4 m/s. Perfectly inelastic: vf = (m₁v₁)/(m₁+m₂).

How much kinetic energy is lost in the inelastic collision above?

The correct answer is: 24 J. ΔKE = KE_initial − KE_final.

Two objects approach each other: m₁=5 kg at 8 m/s and m₂=3 kg at −4 m/s (elastic). Find v₁' and v₂'.

The correct answer is: v₁' = -1 m/s, v₂' = 11 m/s. You can work it out this way: use the elastic collision formulas. Check that total momentum and KE are conserved.

What does a coefficient of restitution of 0.7 mean physically? Calculate post-collision velocities for equal masses.

The correct answer is: The relative speed after is 70% of the relative speed before. E = relative speed after / relative speed before. e=1 is elastic, e=0 is perfectly inelastic.

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Momentum & Collisions

Conservation of momentum in elastic and inelastic collisions

Would you rather be hit by a ping-pong ball or a bowling ball at the same speed?

Your intuition already knows about momentum — mass times velocity determines impact.

How it works

The law of conservation of momentum states that the total momentum of an isolated system remains constant regardless of internal interactions. In elastic collisions, both momentum and kinetic energy are conserved. In perfectly inelastic collisions, objects stick together — momentum is still conserved but kinetic energy is not (it converts to heat/deformation). The coefficient of restitution e ranges from 0 (perfectly inelastic) to 1 (perfectly elastic).

What you'll learn

Conservation of Momentum. In any collision where no external forces act, the total momentum before equals the total momentum after. This holds regardless of whether the collision is elastic, inelastic, or somewhere in between. It is one of the most fundamental conservation laws in physics.
Elastic vs. Inelastic Collisions. In an elastic collision, both momentum and kinetic energy are conserved — billiard balls are a close approximation. In a perfectly inelastic collision, the objects stick together and kinetic energy is converted to heat or deformation. Most real collisions fall somewhere between these extremes.
Impulse and Momentum Change. Impulse is the product of force and the time interval over which it acts. It equals the change in momentum of an object. This is why airbags work: they increase the collision time, reducing the peak force while delivering the same impulse.
The Equal-Mass Trick. When two objects of equal mass collide elastically and one is initially at rest, something remarkable happens: the moving object stops completely and the stationary object moves off with the original velocity. This is Newton's cradle in action.

Step-by-step

Five sliders — Mass m₁, Mass m₂, Velocity v₁, Velocity v₂, and Restitution e.
The sim runs continuously (Space toggles Play/Pause): set your state and watch the spheres close, collide, and separate on the track between two elastic walls.
MOMENTUM DATA prices p₁, p₂, and p_total live, and every impact prints its own receipt — momentum before → after and the KE change.
Slide e from 1 (elastic) through 0.6 (partial) to 0 (the spheres leave together, perfectly inelastic).
Try equal masses with one stationary object — the classic AP question.

Key formulas

$p = mv$ Linear momentum
$p_{total} = m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'$ Conservation of momentum
$v_1' = \frac{(m_1 - m_2)v_1 + 2m_2v_2}{m_1 + m_2}$ Post-collision velocity of object 1 (elastic)
$v_2' = \frac{(m_2 - m_1)v_2 + 2m_1v_1}{m_1 + m_2}$ Post-collision velocity of object 2 (elastic)
$v_f = \frac{m_1v_1 + m_2v_2}{m_1 + m_2}$ Final velocity for perfectly inelastic collision
$J = \Delta p = F\Delta t$ Impulse-momentum theorem

Frequently asked questions

A 3 kg object at 4 m/s hits a stationary 3 kg object elastically. What are the final velocities?: The correct answer is: v₁' = 0 m/s, v₂' = 4 m/s. For equal masses in elastic collision: v₁'=0, v₂'=v₁ (momentum transfers completely).
A 4 kg cart at 6 m/s collides and sticks to a 2 kg cart at rest. Find final velocity.: The correct answer is: 4 m/s. Perfectly inelastic: vf = (m₁v₁)/(m₁+m₂).
How much kinetic energy is lost in the inelastic collision above?: The correct answer is: 24 J. ΔKE = KE_initial − KE_final.
Two objects approach each other: m₁=5 kg at 8 m/s and m₂=3 kg at −4 m/s (elastic). Find v₁' and v₂'.: The correct answer is: v₁' = -1 m/s, v₂' = 11 m/s. You can work it out this way: use the elastic collision formulas. Check that total momentum and KE are conserved.
What does a coefficient of restitution of 0.7 mean physically? Calculate post-collision velocities for equal masses.: The correct answer is: The relative speed after is 70% of the relative speed before. E = relative speed after / relative speed before. e=1 is elastic, e=0 is perfectly inelastic.