What is the law of conservation of momentum?

The law of conservation of momentum states that in a closed system with no external forces, the total momentum before a collision equals the total momentum after the collision. Mathematically: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. This law holds for both elastic and inelastic collisions and is one of the most fundamental principles in physics.

What is the difference between elastic and inelastic collisions?

In an elastic collision, both momentum and kinetic energy are conserved. The objects bounce off each other, and no kinetic energy is lost to heat, sound, or deformation. In an inelastic collision, momentum is conserved but kinetic energy is not fully conserved — some is converted to other forms. A perfectly inelastic collision is one where the objects stick together after impact.

What are the SI units of momentum?

The SI unit of momentum is kg·m/s (kilogram-metre per second). This can also be written as N·s (newton-second), since 1 N = 1 kg·m/s². Both are equivalent and correct. In the CGS system, the unit is g·cm/s. In the imperial system, it is lb·ft/s or slug·ft/s.

Is momentum conserved in every collision?

Momentum is conserved in every collision as long as there are no net external forces acting on the system during the collision. In reality, collisions happen so quickly that external forces like friction and gravity have negligible time to act, so momentum conservation is an excellent approximation. Kinetic energy, however, is only conserved in elastic collisions.

Momentum Calculator – p = mv, Impulse & Collisions

Presets:

Solve for

Mass (m)

Velocity (v)

Momentum (p)

Solve for

Force (F)

Time Δt (s)

Impulse J (N·s = kg·m/s)

Enter masses and initial velocities. Positive = rightward, negative = leftward.

Mass 1 (m₁) — kg

Initial Velocity u₁ (m/s)

Mass 2 (m₂) — kg

Initial Velocity u₂ (m/s)

Objects stick together after collision (perfectly inelastic).

Mass 1 (m₁) — kg

Initial Velocity u₁ (m/s)

Mass 2 (m₂) — kg

Initial Velocity u₂ (m/s)

What Is Momentum?

Momentum (symbol p) is a fundamental quantity in classical mechanics that describes the quantity of motion an object possesses. It depends on two things: how much mass the object has and how fast it is moving. A stationary object has zero momentum; a moving object always has non-zero momentum.

Momentum is a vector quantity — direction matters. A car moving north has opposite momentum to an identical car moving south at the same speed. This sign convention is crucial for collision calculations.

p = mv

Linear momentum

J = FΔt

Impulse = change in momentum

kg·m/s

SI unit of momentum

The Momentum Formula p = mv

Linear momentum is the product of an object's mass and velocity. The formula can be rearranged to solve for any of the three variables:

p = m × v | m = p / v | v = p / m

p — momentum (kg·m/s)

The vector quantity describing the state of motion. Direction follows velocity's direction.

m — mass (kg)

The scalar measure of how much matter the object contains. Always positive.

v — velocity (m/s)

The rate of change of position. Can be positive or negative depending on direction.

Impulse–Momentum Theorem

The impulse-momentum theorem states that the impulse applied to an object equals the change in its momentum:

J = F × Δt = Δp = m(v − u)

This is why car airbags increase the collision time — a longer Δt means a smaller force F for the same change in momentum, reducing injury. Similarly, a cricket bat following through maximises impulse by extending contact time.

Conservation of Momentum

In a closed system with no external forces, total momentum is constant:

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

This holds for all collision types — elastic, inelastic, and explosive. It is a direct consequence of Newton's third law: equal and opposite forces between colliding objects.

Elastic vs Inelastic Collisions

Type	Momentum	Kinetic Energy	Example
Elastic	Conserved	Conserved	Billiard balls, atomic collisions
Inelastic	Conserved	Not fully conserved	Car crash, football tackle

Elastic collision formulas: v₁′ = ((m₁−m₂)u₁ + 2m₂u₂) / (m₁+m₂) and v₂′ = ((m₂−m₁)u₂ + 2m₁u₁) / (m₁+m₂). Inelastic: vf = (m₁u₁ + m₂u₂) / (m₁+m₂).

Momentum in Everyday Life

Momentum is everywhere: rocket propulsion relies on conservation of momentum as exhaust gases are ejected backwards; car safety features (crumple zones, airbags) exploit impulse to reduce peak collision force; sports physics from cricket to American football all depend on momentum transfer.

Rocket thrust

Exhaust gas p = −(gas p) keeps total momentum zero

Airbags

Increase Δt to reduce peak force on passenger

Cricket batting

Follow-through extends contact time → more impulse

Newton's cradle

Near-elastic collisions transfer momentum ball-to-ball

Worked Examples

Example 1 — Momentum of a cricket ball at 145 km/h

Given: m = 0.16 kg, v = 145 km/h = 40.28 m/s

Formula: p = m × v

Step 1: p = 0.16 × 40.28

Answer: p = 6.44 kg·m/s

Example 2 — Impulse from airbag (70 kg driver, 60→0 km/h in 0.05 s)

Given: m = 70 kg, Δv = 16.67 m/s, Δt = 0.05 s

Formula: J = mΔv; F = J/Δt

Step 1: J = 70 × 16.67 = 1,167 N·s

Step 2: F = 1,167 / 0.05

Answer: F = 23,340 N ≈ 23.3 kN

Example 3 — Elastic collision (2 kg at 5 m/s hits 3 kg at rest)

Given: m₁=2, u₁=5, m₂=3, u₂=0

v₁′: ((2−3)×5 + 2×3×0) / (2+3) = −5/5 = −1 m/s

v₂′: ((3−2)×0 + 2×2×5) / (2+3) = 20/5 = 4 m/s

Answer: v₁′ = −1 m/s (bounces back), v₂′ = +4 m/s

Example 4 — Football tackle (90 kg at 8 m/s vs 110 kg at −6 m/s)

Given: m₁=90, u₁=8, m₂=110, u₂=−6

Formula: vf = (m₁u₁ + m₂u₂)/(m₁+m₂)

Step 1: vf = (720 − 660) / 200 = 60/200

Answer: vf = 0.3 m/s (in direction of m₁)

Frequently Asked Questions

Momentum is a measure of the quantity of motion of an object. It is defined as the product of an object's mass and its velocity: p = m × v. Momentum is a vector quantity, meaning it has both magnitude and direction. An object at rest has zero momentum. A heavier or faster-moving object has greater momentum.

The formula p = mv states that linear momentum equals mass multiplied by velocity. In SI units, momentum is measured in kg·m/s. For example, a 1500 kg car travelling at 20 m/s has momentum p = 1500 × 20 = 30,000 kg·m/s. It can be rearranged to find mass (m = p/v) or velocity (v = p/m).

Momentum (p = mv) describes the current state of motion of an object. Impulse (J = FΔt) describes the change in momentum caused by a force applied over a time interval. The impulse-momentum theorem states J = Δp = m(v − u). A large force over a short time or a small force over a long time can produce the same impulse and the same change in momentum.

In a closed system with no external forces, the total momentum before a collision equals the total momentum after: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. This law holds for both elastic and inelastic collisions and is one of the most fundamental principles in physics, following directly from Newton's third law.

In an elastic collision, both momentum and kinetic energy are conserved — the objects bounce off each other perfectly. In an inelastic collision, momentum is conserved but kinetic energy is not — some energy is lost to heat, sound, or deformation. A perfectly inelastic collision is one where the objects stick together after impact, maximising energy loss.

Yes. Momentum is a vector quantity. By convention, if one direction is positive, an object moving in the opposite direction has negative momentum. For example, in a head-on collision, if one car has p = +30,000 kg·m/s, an opposing car might have p = −20,000 kg·m/s. The total system momentum is +10,000 kg·m/s.

The SI unit of momentum is kg·m/s (kilogram-metre per second). This is equivalent to N·s (newton-second), since 1 N = 1 kg·m/s². Both are correct and interchangeable. In CGS the unit is g·cm/s, and in the imperial system lb·ft/s or slug·ft/s.

Momentum is conserved in every collision as long as there are no net external forces on the system during the collision. In practice, collisions happen so quickly that external forces have negligible time to act, so momentum conservation is an excellent approximation. Kinetic energy, however, is only conserved in elastic collisions.