Momentum Formula: p = mv — Definition, Dimensional Formula, Derivation & Solved Examples
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What Is the Momentum Formula?
The momentum formula defines momentum as the product of an object's mass and velocity:
p = mv
where p is momentum (kg·m/s), m is mass (kg), and v is velocity (m/s). Since velocity is a vector, the momentum formula also makes momentum a vector — it always points in the same direction as the object's velocity. A heavier object and a faster object can have identical momentum if their mass × velocity product is equal; the momentum formula in physics depends on both quantities together, never on either alone.
New to this chapter? Start with Class 11 Laws of Motion Notes →
Linear Momentum Formula — Is It the Same as "Momentum"?
Yes. When physics problems say "momentum formula" without qualification, they almost always mean the linear momentum formula, p = mv, as opposed to angular momentum (L = Iω), which applies to rotating bodies. This page covers linear momentum in full.
SI Unit and Dimensional Formula of Momentum
Property | Value |
|---|---|
SI Unit | kg·m/s (equivalently, N·s) |
CGS Unit | g·cm/s |
Dimensional formula of momentum | [M¹L¹T⁻¹] |
Linear momentum dimensional formula | [M¹L¹T⁻¹] (same — "linear" simply clarifies it's not angular momentum) |
Type of Quantity | Vector |
What is the dimensional formula of linear momentum, and how is it derived?
Since p = mv, and mass has dimension [M¹] while velocity has dimension [L¹T⁻¹], multiplying the two gives the momentum dimensional formula: [M¹L¹T⁻¹]. This exact question is a recurring one-mark question in CBSE boards and JEE Main.
Momentum Formula for Class 9, Class 11, and Class 12 — What Changes at Each Level
Class | What's covered | Formula |
|---|---|---|
Class 9 (NCERT, Force and Laws of Motion) | Basic definition and units only | p = mv |
Class 11 (Laws of Motion) | Full vector treatment, impulse-momentum theorem, conservation of momentum, collisions | p = mv, F = dp/dt |
Class 12 (contextual use) | Reappears in Electromagnetic Induction, photon momentum, Modern Physics | p = mv, p = h/λ (de Broglie) |
If you're revising for Class 9 boards, only the basic momentum formula and its SI unit matter. JEE Main and NEET aspirants need the full Class 11 treatment below — impulse, collisions, and conservation — since that's where the actual numerical questions come from.
How Is the Momentum Formula Derived from Newton's Second Law?
Newton's second law states that the rate of change of momentum is directly proportional to the net applied force, in the direction of that force:
F = dp/dt
For a constant-mass system, substituting p = mv gives:
F = d(mv)/dt = m(dv/dt) = ma
This shows F = ma is a special case of the more general momentum formula F = dp/dt — the rate of change of momentum formula — which also holds when mass changes, such as in rocket propulsion problems.
Momentum Formula with Force — Quick Reference
Situation | Formula |
|---|---|
Constant mass | F = ma |
Variable mass | F = dp/dt = m(dv/dt) + v(dm/dt) |
Impulsive force | F = Δp/Δt (average force over a short time) |
What Is the Impulse-Momentum Theorem?
Impulse (J) uses the change in momentum formula directly:
J = FΔt = Δp = m(v − u)
The impulse delivered to an object equals its change in momentum — a direct consequence of the momentum formula and Newton's second law. This is why airbags reduce injury: increasing impact time (Δt) reduces average force for the same change in momentum. Practice this pattern with JEE Main Chapterwise PYQ →
Momentum Formula in Terms of Kinetic Energy
p = √(2mKE), or equivalently, KE = p²/2m
Derived by substituting v = p/m into KE = ½mv². This is a frequent JEE Main shortcut for problems where velocity isn't given directly but kinetic energy is.
Law of Conservation of Momentum
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
This conservation of momentum formula is applied extensively to elastic and inelastic collisions. Collision numericals recur throughout JEE Advanced Chapterwise PYQ →
Total Momentum Formula for a System of Particles
For more than two bodies, the total momentum formula generalises to:
P(total) = p₁ + p₂ + p₃ + ... = m₁v₁ + m₂v₂ + m₃v₃ + ...
This is what's actually conserved in any closed system — the two-body version above is the simplest case.
Momentum Formula for Collision — Elastic vs Inelastic
Collision Type | What's Conserved | Formula |
|---|---|---|
Elastic | Momentum and kinetic energy | m₁u₁+m₂u₂ = m₁v₁+m₂v₂, and ½m₁u₁²+½m₂u₂² = ½m₁v₁²+½m₂v₂² |
Perfectly inelastic | Momentum only (bodies stick together) | m₁u₁ + m₂u₂ = (m₁ + m₂)v |
Inelastic (not perfectly) | Momentum only | m₁u₁+m₂u₂ = m₁v₁+m₂v₂, KE is lost |
This distinction is one of the most common error sources in JEE Main numericals — both types conserve momentum, but only elastic collisions conserve kinetic energy too.
What Is the Difference Between Linear Momentum and Angular Momentum?
Feature | Linear Momentum | Angular Momentum |
|---|---|---|
Formula | p = mv | L = Iω (or L = r × p) |
Applies to | Objects moving in a straight line or general translation | Objects rotating about an axis |
SI Unit | kg·m/s | kg·m²/s |
Quantity Type | Vector | Vector (pseudo-vector) |
This page covers linear momentum. For rotational problems, see Angular Momentum Formula →
How Do You Solve Problems Using the Momentum Formula? (Solved Examples)
Example 1: Basic momentum calculation. A ball of mass 0.5 kg moves with a velocity of 10 m/s. Find its momentum. Solution: p = mv = 0.5 × 10 = 5 kg·m/s
Example 2: Momentum change using impulse. A cricket ball of mass 0.15 kg moving at 12 m/s is hit back at 20 m/s in the opposite direction. Find the change in momentum. Solution: Δp = m(v − u) = 0.15 × [20 − (−12)] = 0.15 × 32 = 4.8 kg·m/s
Example 3: Conservation of momentum in collision. A 2 kg object moving at 3 m/s collides with a stationary 1 kg object, and they stick together. Find their common velocity after collision. Solution: Using m₁u₁ + m₂u₂ = (m₁ + m₂)v 2(3) + 1(0) = (2+1)v → 6 = 3v → v = 2 m/s
Example 4: Force from rate of change of momentum. A force acting on a 4 kg object changes its velocity from 2 m/s to 10 m/s in 4 seconds. Find the average force. Solution: F = Δp/Δt = m(v−u)/t = 4(10−2)/4 = 4(8)/4 = 8 N
Why Is Understanding the Momentum Formula Important for JEE and NEET?
Momentum-based questions appear consistently in both JEE Main and NEET, not only within Laws of Motion but also embedded in Work-Energy-Power (collisions), System of Particles (center of mass and angular momentum), and even Rotational Motion. A clear grasp of the momentum formula, impulse-momentum theorem, and conservation of momentum is essential, since these concepts are rarely tested in isolation and often combined with energy conservation in multi-step numerical problems. Test this exact combination with NEET Chapterwise PYQ, since momentum-energy questions appear regularly in NEET Physics too.
How Should Students Practice Momentum Formula Problems?
- Start with direct substitution problems (p = mv) before attempting impulse or collision-based questions.
- Always define a positive direction before solving collision problems, since momentum is a vector and sign errors are the most common mistake.
- Practice both elastic and inelastic collision problems separately, since they use different conservation conditions (only inelastic collisions conserve momentum but not kinetic energy).
- Revisit the impulse-momentum theorem alongside Newton's second law, since exam questions often test the connection between the two rather than either formula in isolation.
Once confident, test recall speed under timed conditions with eSaral's JEE Test Series, which includes a dedicated Laws of Motion mock section.
Which Other eSaral Physics Resources Should Class 11 Students Check Next?
- Class 11 Laws of Motion Notes for JEE & NEET
- NCERT Solutions Class 11 Physics Chapter 4 — Laws of Motion
- What Is Impulse? — Definition, Formula & Solved Examples
- Applications of Impulse — Class 11 Physics
- Linear Momentum Questions and Answers for Class 11 Physics
- Laws of Motion Class 11 Questions with Answers
- Work Energy and Power Class 11 Physics Notes
- CBSE Class 11 Physics Revision Notes with Weightage
- Practice momentum and collision-based JEE Main questions: JEE Main Chapterwise PYQ with Solutions →
Want to master momentum, impulse, and collisions with solved JEE and NEET problems? eSaral's Class 11 Physics course covers Laws of Motion in depth with concept videos by IIT Bombay faculty and chapter-wise practice tests. Explore Class 11 Physics Courses on eSaral →
Frequently Asked Questions
Is momentum a scalar or a vector quantity?
Momentum is a vector quantity because it depends on velocity, which has both magnitude and direction. The direction of an object's momentum is always the same as the direction of its velocity.
What is the SI unit of momentum?
The SI unit of momentum is kilogram-metre per second (kg·m/s), which is equivalent to newton-second (N·s), since momentum can also be expressed as the product of force and time (impulse).
What is the momentum formula?
The momentum formula is p = mv, where p is momentum, m is the mass of the object, and v is its velocity. Momentum is a vector quantity, so it has both magnitude and direction, same as velocity.
What is the law of conservation of momentum?
The law of conservation of momentum states that in a closed system with no external force, total momentum before a collision equals total momentum after the collision: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂.
What is the impulse-momentum theorem?
The impulse-momentum theorem states that impulse equals the change in momentum: J = FΔt = m(v − u). It connects the force applied over a time interval to the resulting change in an object's momentum.
How is the momentum formula related to Newton's second law?
Newton's second law states F = dp/dt, meaning force equals the rate of change of momentum. For constant mass, this simplifies to the familiar F = ma, showing that F = ma is a special case of the more general momentum-based form of Newton's second law.
Why is momentum important in collision problems?
Momentum is important in collision problems because it is always conserved in a closed system, regardless of whether the collision is elastic or inelastic, making the momentum formula the primary tool for finding unknown velocities after a collision.
Does momentum depend only on mass or only on velocity?
Momentum depends on both mass and velocity together, not on either alone. A heavy object moving slowly and a light object moving fast can have the same momentum if the product of their mass and velocity is equal.
What is the dimensional formula of momentum?
The dimensional formula of momentum is [M¹L¹T⁻¹], derived from its formula p = mv, since mass has dimension [M] and velocity has dimension [L¹T⁻¹].
What is the difference between momentum and force?
Momentum (p = mv) measures the quantity of motion an object possesses, while force (F = dp/dt) measures the rate at which that momentum changes. Force causes a change in momentum; momentum itself is a property of the moving object.