ExploreSims
ExploreSims / Physics / Mechanics / Momentum & Collisions
Premium experiment • conservation vs dissipation

PHYSICS • MECHANICS

Momentum & Collisions

Animate a one-dimensional impact, vary both masses and velocities, and separate conserved momentum from restitution-driven kinetic-energy loss.

● COLLISION LABMomentum & Collisions
Final v₁
Final v₂
KE lost
Center-of-mass v

Reasoning inspector

live model state
Momentum before—
Momentum after—
Momentum error—
KE before—
KE after—
KE retained—
Energy lost—
Impulse on object 1—
Impulse on object 2—
Center-of-mass velocity—
Relative speed—
Collision type—

Conservation plus restitution

The two final velocities are determined by total momentum and the coefficient of restitution. Lower restitution preserves momentum while reducing separation speed and kinetic energy.

LEARN WITH THE SIMULATION

Understand Momentum & Collisions by experimenting.

START HERE

A one-dimensional collision can conserve total momentum while losing a large fraction of kinetic energy. This laboratory solves the two final velocities from momentum conservation together with the coefficient of restitution, then exposes momentum before and after, kinetic energy retention, energy loss, equal-and-opposite impulses, center-of-mass velocity and relative approach and separation speeds. The animation lets the algebra be read as an actual before/impact/after event.

BUILD THE IDEA

What you should understand first

  • Use momentum conservation and restitution together to determine both final velocities.
  • Separate momentum conservation from kinetic-energy conservation and quantify the dissipated energy.
  • Relate coefficient of restitution to relative separation speed rather than to either object's individual velocity.
  • Use equal-and-opposite impulse to explain why unequal masses undergo different velocity changes.

TRY IT YOURSELF

How can momentum remain unchanged while kinetic energy falls as restitution decreases?

  1. Choose unequal masses and approaching initial velocities, then record total momentum and initial kinetic energy.
  2. Set restitution near 1 and note final velocities and retained kinetic energy.
  3. Lower restitution in steps while keeping masses and initial velocities fixed.
  4. Watch total momentum remain constant while relative separation speed and final kinetic energy decrease.
Read the visual:

Restitution controls the relative speed after impact, while momentum conservation constrains the center-of-mass motion. Energy associated with motion relative to the center of mass can be converted into deformation, heat, sound or other internal modes without changing total momentum of the isolated two-object system.

WHY IT WORKS 01

Momentum fixes center-of-mass motion through the collision

For an isolated two-object system, total momentum p = m₁u₁ + m₂u₂ equals m₁v₁ + m₂v₂ after impact. Dividing by total mass gives the center-of-mass velocity, which stays unchanged. The collision can dramatically rearrange each object's velocity while leaving this mass-weighted reference motion invariant.

WHY IT WORKS 02

Restitution constrains relative motion, not total motion

The coefficient of restitution is e = (v₂−v₁)/(u₁−u₂) for the sign convention used when object 1 approaches object 2. At e = 1 the relative separation speed equals the approach speed; at e = 0 the final velocities are equal and the objects have no relative separation immediately after the idealized impact. Combining this relation with momentum conservation uniquely determines the two final velocities.

WHY IT WORKS 03

Where kinetic energy can go without violating conservation laws

In an inelastic collision, translational kinetic energy can become deformation, vibration, heat, sound or other internal energy. Total energy is still conserved in a sufficiently complete physical description, but the simple model tracks only macroscopic translational kinetic energy. Momentum can remain exactly conserved at the same time because internal forces occur in equal-and-opposite pairs.

DO NOT MIX THESE UP

Common misconception

Momentum conservation does not imply kinetic-energy conservation. Equal and opposite collision impulses do not require equal velocity changes because Δv = J/m depends on mass. The coefficient of restitution is a ratio of relative separation and approach speeds, not the percentage of total kinetic energy retained; energy retention also depends on masses and center-of-mass motion.

PRACTICE CHALLENGE

Can you hit a target kinetic-energy retention while demonstrating that momentum and collision impulses still obey their own conservation constraints?

  • Choose a target KE-retention percentage in the premium controls.
  • Change restitution until the target is reached while keeping masses and initial velocities fixed.
  • Compare momentum before and after and the two impulse values.
  • Explain why the impulses are equal and opposite even when the two velocity changes are different.
Reveal the reasoning

Restitution changes the relative speed after impact and therefore changes how much translational kinetic energy remains. Total momentum stays fixed for the isolated modeled system, and the collision impulse on object 2 is the negative of the impulse on object 1. Unequal masses convert those equal impulse magnitudes into unequal velocity changes.

MODEL BOUNDARIES

What this simulation assumes

Every simulation simplifies reality or a mathematical system. These are the assumptions and limits to keep in mind while interpreting the results.

Model assumptions

  • The collision is one-dimensional and both objects move along the same straight line.
  • The two-object system is isolated during the short collision interval, so external impulse is neglected.
  • A single coefficient of restitution between 0 and 1 describes the ratio of separation to approach speed.

Model limitations

  • Rotation, deformation, friction with the environment and multidimensional impact geometry are not represented.
  • The restitution coefficient is treated as constant even though real materials can depend on speed, temperature and deformation history.

CONNECT IT

Where this matters

  • Vehicle crash reconstruction
  • Sports-ball and impact analysis
  • Particle and mechanical collision models
  • Understanding impulse and energy dissipation in impacts

WORDS THAT MATTER

Key vocabulary

Momentum
Vector quantity p = mv; total momentum is conserved for the isolated collision modeled here.
Coefficient of restitution
Ratio of relative separation speed after impact to relative approach speed before impact.
Impulse
Change in momentum J = Δp delivered during the collision.
Center-of-mass velocity
Total momentum divided by total mass; invariant here because no external impulse is modeled.
Kinetic-energy retention
Ratio of final to initial translational kinetic energy.