Momentum Calculator

About the Momentum Calculator

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?"

When This Page Helps

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.

How to Use the Inputs

1. Select single-object mode for basic p = mv calculation, or collision mode for two-body analysis.
2. Enter mass and velocity for each object (use negative velocity for opposite direction).
3. Choose elastic or perfectly inelastic collision type.
4. Review post-collision velocities, momentum (always conserved), and kinetic energy.
5. Use the energy bar chart to visualize KE transfer and loss.
6. Reference the collision types table for key properties.

Example Calculation

Tips & Best Practices

• Use negative velocity to indicate motion in the opposite direction.
• In equal-mass elastic collisions, the objects simply exchange velocities.
• Perfectly inelastic collisions lose the maximum possible KE while conserving momentum.
• The center-of-mass velocity is unchanged by any collision: v_cm = (m₁v₁ + m₂v₂)/(m₁+m₂).
• For real collisions, measure the coefficient of restitution to interpolate between elastic and inelastic.

Conservation Laws in Physics

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.

Collision Analysis in Engineering

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.

Momentum in Particle Physics

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.

Frequently Asked Questions

• Why is momentum always conserved?

  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.

• What is the difference between elastic and inelastic collisions?

  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.

• Can kinetic energy increase in a collision?

  In explosions (internal energy release), KE increases while momentum is still conserved. In ordinary collisions, KE either stays the same (elastic) or decreases (inelastic).

• What is the coefficient of restitution?

  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.

• How does this apply to rocket propulsion?

  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.

• What about angular momentum?

  Angular momentum (L = Iω or L = r × p) is separately conserved in the absence of external torques. This calculator handles linear momentum only.