What is linear momentum in physics and classical mechanics?

Linear momentum (denoted as p) is a fundamental vector quantity representing the quantity of motion an object possesses, calculated as the product of its mass and velocity: p = m · v.

What is the mathematical formula for linear momentum and its SI units?

The classical momentum formula is p = m · v, where m is mass in kilograms (kg) and v is velocity in meters per second (m/s). The standard SI unit of momentum is kilogram-meter per second (kg·m/s), which is dimensionally equivalent to Newton-seconds (N·s).

How is momentum related to kinetic energy (E_k)?

Momentum and kinetic energy are related by: E_k = p² / (2m) or p = √(2m · E_k). While momentum is a vector (p = mv) conserved in all collisions, kinetic energy is a scalar (E_k = 1/2 mv²) that is only conserved in perfectly elastic collisions.

What is the impulse-momentum theorem and how does it calculate impact force?

The impulse-momentum theorem states that the impulse (J) applied by a net force equals the change in momentum: J = Δp = m(v_f - v_i) = F_avg · Δt. The average impact force is: F_avg = Δp / Δt.

What is the law of conservation of momentum?

In any closed, isolated physical system with no external forces (Σ F_ext = 0), the total vector momentum before an interaction equals the total vector momentum after: Σ p_initial = Σ p_final.

What is the difference between elastic and inelastic collisions?

In an elastic collision, both total momentum and total kinetic energy are conserved (no energy lost to heat or sound). In an inelastic collision, total momentum is conserved, but kinetic energy is converted into heat, deformation, or sound. In a perfectly inelastic collision, the two colliding bodies stick together.

How do you calculate post-collision velocities in a 1D elastic collision?

For two masses m₁ and m₂ with initial velocities v₁ and v₂: v₁' = [(m₁ - m₂) / (m₁ + m₂)] · v₁ + [2m₂ / (m₁ + m₂)] · v₂ and v₂' = [2m₁ / (m₁ + m₂)] · v₁ + [(m₂ - m₁) / (m₁ + m₂)] · v₂.

What is relativistic momentum and when does classical momentum break down?

As an object's velocity approaches the speed of light (v → c), classical mechanics fails and special relativity applies: p_rel = γ · m · v = (m · v) / √(1 - v²/c²), where γ is the Lorentz factor.

What is de Broglie wavelength and how is quantum matter momentum calculated?

In quantum mechanics, every moving particle exhibits wave-particle duality with de Broglie wavelength λ = h / p, where h is Planck's constant (6.626 × 10⁻³⁴ J·s) and p is momentum.

How do automotive engineers, physicists, and sports analysts use the Momentum Calculator?

Automotive engineers calculate crash crumple zone impact forces and airbag deployment timing, particle physicists track high-energy subatomic collisions, and sports scientists optimize bat swing momentum and ball rebound velocities.

⚡ 100% Free • Linear Momentum, Impulse, Collisions & Kinetic Energy Solver

Momentum Calculator

Calculate linear momentum (p = m·v), relativistic momentum, impulse (J = F·Δt), kinetic energy (Ek = p²/2m), and 1D elastic/inelastic collisions with the free Momentum Calculator.

Physics Presets:

Calculation Method

Object Mass (m) 1,500 kg (3,307 lbs)

Velocity (v) 25.0 m/s (90.0 km/h)

Impulse & Impact Time (Optional) Δt = 0.100 s

Collision / Stopping Duration (Δt in seconds)

Momentum (p) 37,500 kg·m/s

Kinetic Energy (E_k) 468,750 J

Calculated Linear Momentum (p)

37,500 kg·m/s

37,500 N·s • 27,651 lb·ft/s • Ek: 468,750 J • Force @ 0.1s: 375.0 kN • λ: 1.178 × 10⁻³⁸ m

Kinetic Energy

468,750 J

345,732 ft·lbf

Impact Force (F_avg)

375.0 kN

84,303 lbf

Imperial Momentum

27,651 lb·ft/s

859.4 slug·ft/s

Lorentz Factor (γ)

1.000000

Relativistic scale

De Broglie (λ)

1.18 × 10⁻³⁸ m

Quantum wavelength

Equivalent Velocity

25.0 m/s

90.0 km/h (55.9 mph)

Step-by-Step Physics & Momentum Derivation

What is Linear Momentum? Physical Definition & Invariant Properties

In classical physics, engineering mechanics, and Newtonian kinematics, linear momentum (symbolized as $\vec{p}$, derived from the Latin petere, meaning "to go towards") is a fundamental vector quantity that quantifies the total quantity of motion possessed by a mass in motion:

1. Product of Mass & Velocity

Linear momentum scales proportionally with both inertia and speed: $\vec{p} = m \vec{v}$. A massive freight train moving slowly possesses equal momentum to a lightweight bullet moving at supersonic speed.

2. Vector Directionality

Momentum is intrinsically directional, possessing both magnitude and spatial orientation along the object's velocity vector, enabling vector summation in multi-dimensional collision analysis.

3. Universal Conservation

According to Noether's theorem, the law of conservation of linear momentum is a direct mathematical consequence of the continuous translational symmetry of space.

The Core Mathematical Formulas for Momentum, Impulse & Kinetic Energy

Summary of the foundational equations used in mechanics, automotive safety, and physics:

Physical Metric	Mathematical Formula	Standard SI Units
Linear Momentum ($p$)	$$p = m \cdot v$$	$\text{kg}\cdot\text{m/s} \equiv \text{N}\cdot\text{s}$
Kinetic Energy from Momentum ($E_k$)	$$E_k = \frac{p^2}{2m} = \frac{1}{2} m v^2$$	$\text{Joules (J)} \equiv \text{N}\cdot\text{m}$
Impulse-Momentum Theorem ($J$)	$$J = \Delta p = F_{\text{avg}} \cdot \Delta t$$	$\text{N}\cdot\text{s}$
Average Impact Force ($F_{\text{avg}}$)	$$F_{\text{avg}} = \frac{\Delta p}{\Delta t} = \frac{m(v_f - v_i)}{\Delta t}$$	$\text{Newtons (N)}$
Relativistic Momentum ($p_{\text{rel}}$)	$$p_{\text{rel}} = \gamma m v = \frac{m v}{\sqrt{1 - v^2/c^2}}$$	$\text{kg}\cdot\text{m/s}$
De Broglie Matter Wavelength ($\lambda$)	$$\lambda = \frac{h}{p} = \frac{h}{m v}$$	$\text{meters (m)}$

How to Use the Momentum Calculator

1 Select Calculation Method

Choose between Standard Momentum ($p = mv$), Velocity Solver ($v = p/m$), Mass Solver ($m = p/v$), or 1D Collision Simulator.

2 Enter Mass & Velocity Values

Input mass in $\text{kg}$, $\text{g}$, $\text{lbs}$, or $\text{tonnes}$ and velocity in $\text{m/s}$, $\text{km/h}$, $\text{ft/s}$, $\text{mph}$, or $c$.

3 Optionally Enter Impact Time ($\Delta t$)

Specify crash or stopping duration to evaluate peak deceleration and average impact force in $\text{kN}$ and $\text{lbf}$.

4 Review Kinetic Energy & Step-by-Step Proof

Inspect linear momentum, kinetic energy, de Broglie matter wavelength, relativistic factor, and live KaTeX derivations.

Momentum ($p = mv$) vs. Kinetic Energy ($E_k = \frac{1}{2}mv^2$): Key Differences

Why momentum and kinetic energy govern fundamentally different physical phenomena:

Linear Momentum ($p = mv$)

A vector quantity that measures resistance to stopping. Strictly conserved in all collisions (elastic and inelastic) regardless of internal friction or thermal energy conversion.

Kinetic Energy ($E_k = \frac{1}{2}mv^2$)

A scalar quantity that measures the mechanical work capacity. Scaled quadratically with velocity ($v^2$) and only conserved in perfectly elastic collisions.

The Impulse-Momentum Theorem: How Crumple Zones & Airbags Save Lives

Applying Newton's second law in its original momentum-rate formulation ($F = \frac{dp}{dt}$):

When bringing an object of momentum $p$ to a complete stop ($\Delta p = p$), the required impulse is constant. Rearranging the impulse formula demonstrates that average impact force is inversely proportional to collision time:

$$F_{\text{avg}} = \frac{\Delta p}{\Delta t} = \frac{m \cdot v_0}{\Delta t}$$

Automotive safety engineers design front crumple zones and deploying airbags to lengthen impact duration from $0.01\text{ s}$ (rigid barrier) to $0.10\text{ s}$, slashing peak deceleration forces on passengers by $90\%$.

Key Features of the Momentum Calculator

Multi-Variable Algebraic Solver

Solve for momentum ($p$), mass ($m$), or velocity ($v$) with automatic unit conversions.

1D Collision Simulator

Simulate two-body elastic, inelastic, and partially inelastic collisions with coefficient of restitution ($e$).

Impulse & Impact Force Analysis

Evaluates stopping time ($\Delta t$) to calculate average impact force in kiloNewtons and pounds-force.

Relativistic Momentum Solver

Applies Einstein's Lorentz factor ($\gamma$) for near-light-speed particle physics calculations.

De Broglie Quantum Wavelength

Computes quantum matter wavelength ($\lambda = h/p$) across macro and atomic scales.

100% In-Browser & Private

Executes instantly on your device with zero server latency and total calculation privacy.

Problems This Momentum Calculator Solves

Eliminates Multi-Unit Conversion Confusion

Prevents dimensional errors when converting between metric ($\text{kg}\cdot\text{m/s}$, $\text{N}\cdot\text{s}$) and imperial ($\text{lb}\cdot\text{ft/s}$, $\text{slug}\cdot\text{ft/s}$) momentum units.

Accurately Models Post-Collision Dynamics

Solves coupled simultaneous momentum and kinetic energy conservation equations for two colliding vehicles or billiard balls without tedious manual algebra.

1D Two-Body Collisions: Elastic, Inelastic & Coefficient of Restitution ($e$)

How energy conservation dictates post-collision rebound velocities:

The general post-collision velocities for two bodies ($m_1, m_2$) with coefficient of restitution $e = \frac{v_2' - v_1'}{v_1 - v_2}$ are:

$$v_1' = \frac{m_1 v_1 + m_2 v_2 - m_2 e (v_1 - v_2)}{m_1 + m_2} \qquad v_2' = \frac{m_1 v_1 + m_2 v_2 + m_1 e (v_1 - v_2)}{m_1 + m_2}$$

Special Relativity: Relativistic Momentum ($p = \gamma mv$)

Why classical momentum fails as velocity approaches the speed of light ($c \approx 3 \times 10^8\text{ m/s}$):

In special relativity, as $v \to c$, the Lorentz factor $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \to \infty$. An infinite force would be required to accelerate any massive body to the speed of light, establishing $c$ as the absolute cosmic speed limit.

Linear Momentum ($p = mv$) vs. Angular Momentum ($L = I\omega$)

Comparing translational motion with rotational orbital dynamics:

Linear Momentum ($\vec{p} = m \vec{v}$)

Translational quantity of motion along a straight-line vector. Changes only when an external net force applies an impulse ($\vec{F} = \frac{d\vec{p}}{dt}$).

Angular Momentum ($\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$)

Rotational quantity of motion about an axis. Governs gyroscopic stabilization, spinning figure skaters, and Keplerian orbital velocity variations ($r_1 v_1 = r_2 v_2$).

Quantum Mechanics: De Broglie Matter Wavelengths & Photon Momentum

Wave-particle duality in quantum physics and solar radiation pressure:

Louis de Broglie postulated that every moving particle possesses an associated quantum wave packet with wavelength inversely proportional to its momentum:

$$\lambda = \frac{h}{p} = \frac{h}{m v} \qquad p_{\text{photon}} = \frac{E}{c} = \frac{h}{\lambda}$$

Because macro objects have immense mass, their de Broglie wavelengths are imperceptibly small ($\sim 10^{-38}\text{ m}$), whereas high-speed electrons possess wavelengths comparable to atomic lattice spacings, powering transmission electron microscopes (TEM).

Rocket Propulsion: Variable-Mass Momentum & The Tsiolkovsky Equation

How continuous mass ejection drives aerospace propulsion in a vacuum:

In deep space with no external air to push against, a rocket accelerates by ejecting high-velocity combustion gases rearward. Conservation of momentum across differential mass elements ($m \, dv = -v_e \, dm$) yields the ideal rocket equation:

$$\Delta v = v_e \ln\left(\frac{m_0}{m_f}\right) = I_{\text{sp}} \cdot g_0 \ln\left(\frac{m_0}{m_f}\right)$$

Center-of-Mass Momentum & Invariant Reference Frames

Why total system momentum equals the motion of its center of mass:

For any multi-particle system or complex vehicle collision, the total momentum equals total system mass multiplied by the velocity of the center of mass ($\vec{v}_{\text{cm}}$):

$$\vec{P}_{\text{total}} = M_{\text{total}} \cdot \vec{v}_{\text{cm}} = \sum m_i \vec{v}_i \qquad \vec{v}_{\text{cm}} = \frac{\sum m_i \vec{v}_i}{\sum m_i} = \text{constant}$$

Because internal collision forces cancel out in pairs (Newton's third law), $\vec{v}_{\text{cm}}$ remains completely unchanged before, during, and after any impact or explosive separation.

The Ballistic Pendulum: Measuring Bullet Speed via Momentum Transfer

Classic experimental apparatus linking momentum conservation with potential energy:

When a high-speed bullet of mass $m$ embeds into a suspended wooden block of mass $M$, momentum is conserved during impact ($m v_0 = (m + M) V$), after which mechanical energy conserves as the pendulum swings up to vertical height $h$:

$$v_0 = \left(\frac{m + M}{m}\right) \sqrt{2 g h}$$

2D Oblique Collisions & The 90° Orthogonal Scattering Angle

Glancing impacts in subatomic physics and billiard mechanics:

In two-dimensional glancing collisions, vector momentum is conserved independently along orthogonal $x$ and $y$ axes:

$$\sum p_{ix} = \sum p_{fx} \qquad \sum p_{iy} = \sum p_{fy}$$

In an elastic collision between two identical masses ($m_1 = m_2$) where one is initially stationary, the outgoing trajectory angles always form a right angle ($\theta_1 + \theta_2 = 90^\circ$), a property exploited by competitive pool players and particle physicists alike.

Frequently Asked Questions

Comprehensive answers to common questions about momentum formulas, impulse calculations, kinetic energy relationships, and collision mechanics.

Physical Metric	Mathematical Formula	Standard SI Units
Linear Momentum (\(p\))	$$p = m \cdot v$$	\(\text{kg}\cdot\text{m/s} \equiv \text{N}\cdot\text{s}\)
Kinetic Energy from Momentum (\(E_k\))	$$E_k = \frac{p^2}{2m} = \frac{1}{2} m v^2$$	\(\text{Joules (J)} \equiv \text{N}\cdot\text{m}\)
Impulse-Momentum Theorem (\(J\))	$$J = \Delta p = F_{\text{avg}} \cdot \Delta t$$	\(\text{N}\cdot\text{s}\)
Average Impact Force (\(F_{\text{avg}}\))	$$F_{\text{avg}} = \frac{\Delta p}{\Delta t} = \frac{m(v_f - v_i)}{\Delta t}$$	\(\text{Newtons (N)}\)
Relativistic Momentum (\(p_{\text{rel}}\))	$$p_{\text{rel}} = \gamma m v = \frac{m v}{\sqrt{1 - v^2/c^2}}$$	\(\text{kg}\cdot\text{m/s}\)
De Broglie Matter Wavelength (\(\lambda\))	$$\lambda = \frac{h}{p} = \frac{h}{m v}$$	\(\text{meters (m)}\)