What is the law of conservation of momentum in physics?

The law of conservation of momentum states that the total linear momentum of an isolated physical system remains constant if no net external forces act on it: Σ p_initial = Σ p_final, or m1·u1 + m2·u2 = m1·v1 + m2·v2.

What is the difference between an elastic collision and an inelastic collision?

In an elastic collision, both total linear momentum and total kinetic energy are conserved (zero kinetic energy lost to heat or deformation). In an inelastic collision, momentum is conserved, but a portion of kinetic energy transforms into internal heat, sound, or structural deformation.

What happens in a perfectly inelastic collision?

In a perfectly inelastic collision, the colliding bodies stick together upon impact and move with a shared common final velocity: vf = (m1·u1 + m2·u2) / (m1 + m2), maximizing kinetic energy dissipation.

What are the formulas for final velocities in a 1D elastic collision?

For two masses m1 and m2 with initial velocities u1 and u2: v1 = ((m1 - m2)/(m1 + m2))·u1 + ((2·m2)/(m1 + m2))·u2, and v2 = ((2·m1)/(m1 + m2))·u1 + ((m2 - m1)/(m1 + m2))·u2.

What is the coefficient of restitution (e) in collision mechanics?

The coefficient of restitution (e) measures collision elasticity as the ratio of relative separation speed to relative approach speed: e = (v2 - v1) / (u1 - u2). An ideal elastic collision has e = 1.0; a perfectly inelastic collision has e = 0.

How do you calculate firearm or rocket recoil velocity using conservation of momentum?

Starting from rest (total initial momentum = 0), gun recoil velocity is: v_gun = -(m_bullet · v_bullet) / m_gun, where the negative sign denotes opposite rearward direction.

Is momentum conserved in two-dimensional (2D) and three-dimensional (3D) collisions?

Yes. Momentum is a vector quantity and is conserved independently along every orthogonal spatial axis: Σ px,initial = Σ px,final, and Σ py,initial = Σ py,final.

Why do billiard balls of equal mass exchange velocities during an elastic head-on collision?

When m1 = m2 in the elastic collision equations, the formulas simplify to v1 = u2 and v2 = u1. The moving ball stops completely while the target ball moves off with the incoming ball's exact initial speed.

How do you calculate kinetic energy loss in an inelastic impact?

Kinetic energy loss is: ΔEk = (1/2·m1·u1² + 1/2·m2·u2²) - (1/2·m1·v1² + 1/2·m2·v2²). For perfectly inelastic impacts, ΔEk = 1/2 · ((m1·m2)/(m1 + m2)) · (u1 - u2)².

How do automotive crash investigators and aerospace engineers use conservation of momentum?

Crash reconstructionists calculate pre-impact vehicle speeds from post-collision wreckage trajectories, and rocket engineers use momentum conservation (Tsiolkovsky rocket equation) to calculate delta-v from propellant exhaust momentum.

⚡ 100% Free • Elastic, Inelastic & Recoil Collision Solver

Conservation of Momentum Calculator

Calculate final collision velocities (m1·u1 + m2·u2 = m1·v1 + m2·v2), momentum (p = m·v), kinetic energy loss, and firearm recoil velocities with the free Conservation of Momentum Calculator.

Collision Physics Presets:

Collision Regime

Body 1 (Target / Rifle)

Mass ($m_1$) 0.17 kg (0.37 lbs)

Initial Velocity ($u_1$) 3.00 m/s (6.7 mph)

Body 2 (Impact Object / Bullet)

Mass ($m_2$) 0.17 kg (0.37 lbs)

Initial Velocity ($u_2$) 0.00 m/s (0.0 mph)

Body 1 Final Velocity ($v_1$) 0.00 m/s

Body 2 Final Velocity ($v_2$) 3.00 m/s

Post-Collision Final Velocities

v₁ = 0.0 m/s • v₂ = 3.0 m/s

Total Momentum: 0.510 kg·m/s (Conserved Δp = 0) • KE Conserved

Final Velocity ($v_1$)

0.00 m/s

0.0 mph

Final Velocity ($v_2$)

3.00 m/s

6.7 mph

Total Momentum ($p$)

0.510 kg·m/s

Strictly Conserved

Kinetic Energy Loss

0.000 J

0.0% loss (Elastic)

Impulse Transferred

0.510 N·s

$J = \Delta p_1 = -\Delta p_2$

Velocity Ratio

1 : 1

Equal mass exchange

Step-by-Step Conservation of Momentum & Kinetic Energy Derivation

What is Conservation of Momentum in Classical Physics?

In classical Newtonian mechanics, the Law of Conservation of Linear Momentum is one of the most fundamental conservation laws in the universe. It dictates that for any closed, isolated physical system subjected to zero net external forces ($\sum \vec{F}_{\text{ext}} = 0$), the total vector momentum before any interaction or collision is strictly equal to the total vector momentum after the interaction:

$$\sum \vec{p}_{\text{initial}} = \sum \vec{p}_{\text{final}} \implies m_1 \vec{u}_1 + m_2 \vec{u}_2 = m_1 \vec{v}_1 + m_2 \vec{v}_2$$

This universal principle derives directly from Newton's Third Law of Motion (for every action, there is an equal and opposite reaction: $\vec{F}_{12} = -\vec{F}_{21}$). During impact, the internal collision forces exchanged between the two bodies are identical in magnitude and opposite in direction over the contact duration $\Delta t$, resulting in equal and opposite momentum changes ($\Delta \vec{p}_1 = -\Delta \vec{p}_2$) with zero net change in total system momentum.

Collision Taxonomy: Elastic, Inelastic, Restitution & Recoil Formulas

Collision Type	Governing Equations	Kinetic Energy Behavior
1D Perfectly Elastic	$$v_1 = \frac{m_1 - m_2}{m_1 + m_2}u_1 + \frac{2m_2}{m_1 + m_2}u_2$$	Strictly Conserved ($\Delta E_k = 0, e = 1.0$)
1D Perfectly Inelastic	$$v_f = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}$$	Maximum Kinetic Energy Loss ($e = 0$)
Coefficient of Restitution	$$v_2 - v_1 = e (u_1 - u_2)$$	Partial Dissipation ($0 < e < 1$)
Firearm / Rocket Recoil	$$v_{\text{recoil}} = -\frac{m_{\text{bullet}}}{m_{\text{gun}}} v_{\text{muzzle}}$$	Chemical Potential $\to$ Kinetic Energy

How to Use the Conservation of Momentum Calculator

1 Select Collision Regime

Choose between 1D Elastic Collision, Perfectly Inelastic (stick together), Coefficient of Restitution ($e$), or Firearm Recoil.

2 Enter Masses & Initial Velocities

Input object masses in kg, grams, or lbs, and velocities with appropriate positive/negative signs for directional movement.

3 Review Post-Collision Velocities

Inspect resulting final speeds ($v_1, v_2$), common merged velocity, or firearm rearward recoil speed.

4 Inspect Momentum & Energy Conservation Proofs

Verify system momentum invariance ($\Delta p = 0$), kinetic energy loss Joules, and step-by-step KaTeX mathematical derivations.

Problems Solved by the Conservation of Momentum Calculator

1. Crash Investigation & Forensics

Reconstructs pre-impact vehicular speeds from post-collision skid trajectories and vehicle mass ratios in forensic accident analysis.

2. Firearm & Artillery Recoil Sizing

Calculates weapon rearward impulse and recoil velocity to size hydraulic recoil dampening springs and muzzle brakes.

3. Physics Laboratory Verification

Verifies linear air track and ballistic pendulum experiments with exact algebraic and kinetic energy balance equations.

Key Features of the Conservation of Momentum Calculator

4 Universal Collision Modes

Seamlessly solves elastic collisions, perfectly inelastic mergers, partial restitution impacts, and explosion recoil.

Kinetic Energy Dissipation Tracker

Quantifies exact Joules and percentage of kinetic energy converted into heat and plastic deformation during inelastic impacts.

Multi-Unit Physics Engine

Supports seamless mass conversions (kg, g, lbs, metric tons) and velocity conversions (m/s, mph, km/h, ft/s).

Rocket Propulsion & Continuous Momentum Conservation

How momentum conservation propels spacecraft in the vacuum of space:

A rocket engine operates by continuously expelling high-velocity propellant exhaust mass ($dm$) rearward at exhaust velocity $v_e$. By conservation of momentum, the forward momentum gained by the spacecraft exactly balances the rearward momentum carried by the exhaust gas. Integrating this continuous differential momentum balance yields the famed Tsiolkovsky Rocket Equation:

$$\Delta v = v_e \ln\left(\frac{m_{\text{initial}}}{m_{\text{final}}}\right)$$

This confirms that rockets do not need air or a physical surface to "push against"; their forward thrust is purely an internal reaction generated by the conservation of linear momentum.

Center of Mass Reference Frame: The Invariance of System Velocity

Why viewing collisions from the center of mass simplifies complex physics:

Regardless of whether a collision is elastic, inelastic, or explosive, the velocity of the system's Center of Mass (CM) remains strictly constant:

$$v_{\text{cm}} = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2} = \frac{p_{\text{total}}}{M_{\text{total}}}$$

In the center of mass reference frame, total momentum is identically zero ($\sum \vec{p}_{\text{cm}} = 0$). In an elastic collision viewed from the CM frame, the bodies simply rebound in reverse directions with unchanged speeds ($u_{1,\text{cm}}' = -u_{1,\text{cm}}$ and $u_{2,\text{cm}}' = -u_{2,\text{cm}}$).

2D Glancing Collisions & Subatomic Particle Physics

How subatomic particle colliders discover new physics through missing transverse momentum:

In two-dimensional glancing collisions (such as angled billiard shots or particle scatterings), linear momentum is conserved independently along both orthogonal spatial axes:

$$m_1 u_1 = m_1 v_1 \cos\theta_1 + m_2 v_2 \cos\theta_2 \qquad 0 = m_1 v_1 \sin\theta_1 - m_2 v_2 \sin\theta_2$$

At high-energy particle colliders like the CERN Large Hadron Collider (LHC), protons collide head-on at nearly the speed of light. Physicists track the vector momentum sum of all debris particles. If the reconstructed transverse momentum does not sum to zero ($\sum \vec{p}_T \neq 0$), the "missing transverse energy" indicates the creation of elusive non-interacting particles, such as neutrinos or hypothetical dark matter particles.

Frequently Asked Questions

Comprehensive answers to common questions about momentum conservation, elastic vs. inelastic collisions, firearm recoil velocities, and kinetic energy loss.

Collision Type	Governing Equations	Kinetic Energy Behavior
1D Perfectly Elastic	$$v_1 = \frac{m_1 - m_2}{m_1 + m_2}u_1 + \frac{2m_2}{m_1 + m_2}u_2$$	Strictly Conserved (\(\Delta E_k = 0, e = 1.0\))
1D Perfectly Inelastic	$$v_f = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}$$	Maximum Kinetic Energy Loss (\(e = 0\))
Coefficient of Restitution	$$v_2 - v_1 = e (u_1 - u_2)$$	Partial Dissipation (\(0 < e < 1\))
Firearm / Rocket Recoil	$$v_{\text{recoil}} = -\frac{m_{\text{bullet}}}{m_{\text{gun}}} v_{\text{muzzle}}$$	Chemical Potential \(\to\) Kinetic Energy