Conservation of Momentum Calculator

Solve collision problems with momentum conservation

Supports unknown/partial cases plus elastic and perfectly inelastic modes

Last updated: December 17, 2025

CreatorFrank Zhao

Collision Type

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Object 1

Mass (m₁)

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Initial velocity (u₁)

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m/s

Final velocity (v₁)

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m/s

Object 2

Mass (m₂)

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Initial velocity (u₂)

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m/s

Final velocity (v₂)

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m/s

Kinetic Energy

Loss of kinetic energy

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Total KE before collision

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Total KE after collision

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Quick Navigation

This guide explains the law of conservation of momentum in plain English, how elastic vs. inelastic collisions differ, and how to use the calculator to solve real two-object collision problems. If you want a quick refresher on what momentum is, our Momentum Calculator is a great companion.

Introduction / Overview
Law of conservation of momentum
Elastic and inelastic collisions
How to use (step-by-step)
Real-world examples
Tips & best practices
Calculation method (formulas)
Related concepts
FAQs
Limitations & sources

Introduction / Overview

The Conservation of Momentum Calculator helps you solve two-object collision problems: given masses and some before/after velocities, it finds the missing values using the momentum conservation equation. It also computes the system’s kinetic energy before and after the collision so you can quickly tell whether energy was lost (or gained).

Practical takeaway: momentum is about “how hard something is moving.” In many collisions, momentum is conserved even when kinetic energy is not.

Who is this for?

Students, teachers, and engineers who need quick collision calculations for homework, lab checks, simulations, or sanity checks.

Why trust the result?

The calculator works in consistent SI base units internally and converts back to your selected units, which helps avoid unit mistakes.

Want to explore energy more directly? Pair this with our Kinetic Energy Calculator or Potential Energy Calculator.

Law of conservation of momentum

In an isolated system (no net external force), the total linear momentum stays constant. For two objects moving along one line, that idea becomes one compact equation:

m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2

Key idea

Momentum doesn’t “disappear.” If one object loses momentum, the other gains it — unless something outside the system (like friction or a push) adds or removes momentum.

A classic mental picture: two toy cars on a nearly frictionless surface. One car hits the other, slows down, and the second speeds up. That speed-up is momentum being transferred.

Elastic and inelastic collisions

Momentum conservation applies to all collision types (as long as the system is isolated). The big difference between collision types is what happens to kinetic energy.

Perfectly elastic

Momentum and kinetic energy are both conserved. Objects bounce apart (think billiard balls).

Partially elastic

Momentum is conserved, but kinetic energy decreases because some energy becomes heat, sound, or deformation.

Perfectly inelastic

Objects stick together and move with a shared final speed. Momentum is conserved; kinetic energy drops.

Tip:

Even if kinetic energy changes, total energy is still conserved — the “missing” kinetic energy usually becomes internal energy (heat, sound, permanent deformation).

How to use the conservation of momentum calculator

The calculator supports multiple collision modes. A quick rule of thumb: Unknown/Partial solves the momentum equation only, while Elastic enforces kinetic energy conservation, and Inelastic enforces a shared final speed.

Enter the masses

Example: $m_1 = 8\text{ kg}$ , $m_2 = 4\text{ kg}$ .

Set the initial velocities

Example: $u_1 = 10\text{ m/s}$ , $u_2 = 0\text{ m/s}$ .

Enter one final velocity (or leave it blank)

Example: $v_1 = 4\text{ m/s}$ (the first object slows down).

Let the calculator solve the missing value

Initial momentum:

8 \times 10 + 4 \times 0 = 80

Final momentum 1:

8 \times 4 = 32

Result:

v_2 = (80 - 32) / 4 = 12\text{ m/s}

Read the kinetic energy summary

The KE panel shows before vs. after energy and the % loss. This helps you interpret how “elastic” the collision was.

Real-world examples / use cases

1) Billiards-style collision

Inputs: $m_1=0.17\text{kg}$ , $m_2=0.17\text{kg}$ , $u_1=2.0\text{m/s}$ , $u_2=0$ .
Result: $v_1 \approx 0$ and $v_2 \approx 2.0\text{m/s}$ (they swap speeds).

2) Car bump (partially elastic)

Inputs: $m_1=1200\text{kg}$ , $m_2=1400\text{kg}$ , $u_1=6\text{m/s}$ , $u_2=0$ .
Interpretation: momentum balances out, but KE loss can be large due to deformation.

3) Bullet into block

Inputs: $m_1=0.01\text{kg}$ , $u_1=400\text{m/s}$ ; $m_2=2.0\text{kg}$ , $u_2=0$ .
Mode: Inelastic → the final speed is shared and much smaller than 400 m/s.

4) “Explosion” scenario

Context: Kinetic energy is added from an internal source.
Use it for: modeling a spring release or internal energy source where $KE_{final} > KE_{initial}$ .

If you’re analyzing just one object’s momentum before/after, our Momentum Calculator can help you break it down.

Tips & best practices

Pro tip:

Use the sign of velocity to describe direction. A negative velocity simply means “moving left” relative to your chosen +x axis.

Pick the right mode: Elastic enforces KE conservation; Inelastic enforces a shared final speed; Unknown/Partial solves momentum only.
Watch the KE panel: if KE increases, you’re modeling an energy-adding event (like a spring release) rather than a passive collision.
Sanity check extremes: if one mass is huge compared to the other, the heavier object’s speed changes less.
Units matter: keep mass and velocity units consistent. The calculator converts internally, but your inputs should represent the same physical setup.

Calculation method / formula explanation

The calculator uses momentum conservation in all modes, and adds extra constraints depending on the collision type. Kinetic energy is computed as a read-only diagnostic.

Core equations

Momentum Conservation

m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2

Kinetic Energy (per object)

KE = \frac{1}{2}mv^2

Mode-specific constraints

Elastic: $KE_{total}$ is conserved. This creates a system of two equations to solve for $v_1$ and $v_2$ .
Inelastic: $v_1 = v_2$ . The objects move together as a single unit after impact.
Unknown / Partial: Uses the momentum equation only; kinetic energy is allowed to vary.

Related concepts / background info

Momentum vs. kinetic energy

Momentum scales linearly with velocity ( $mv$ ), while kinetic energy scales with the square of velocity ( $mv^2$ ). That squared term is why KE losses are often much more sensitive to speed changes than momentum.

What “isolated system” really means

In practice, isolation means external forces like friction or gravity are minor during the very short timeframe of the collision itself.

If you’re looking at energy budgets (rather than momentum), our Kinetic Energy Calculator can help you compute energy per object.

Frequently asked questions (FAQs)

What is the principle of conservation of momentum?

If the net external force on a system is zero (or negligible during the collision), the system’s total linear momentum stays constant.

When is momentum conserved in real life?

Momentum is conserved when external impulses are small compared to the collision impulse — for example, short impacts on low-friction surfaces.

Can kinetic energy increase after a collision?

It can in special cases (e.g., explosions, spring release, internal energy sources). In everyday passive collisions, $KE$ usually decreases. The calculator highlights $KE$ increases so you can interpret the scenario correctly.

Why do elastic collisions feel “rare”?

Perfect elasticity is an idealization. Real materials deform and produce heat/sound. Hard, smooth objects (like steel balls) can get close.

What makes a rocket move?

A rocket moves because the exhaust gases are pushed backward, and the rocket gains equal and opposite momentum forward. That’s conservation of linear momentum in action.

Do I need to enter kinetic energy values?

No — the kinetic energy section is calculated from masses and velocities and shown as a helpful diagnostic.

Limitations / disclaimers & external references

Limitations

This calculator assumes 1D motion (everything along one line).
External forces (friction, pushing, gravity components along the axis) are not modeled explicitly.
Real collisions can involve rotation, spin, and deformation; results are best used as an idealized estimate.

Note:

This tool is for educational use and should not replace professional engineering judgment for safety-critical designs.

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