Momentum Calculator

Calculate momentum, mass, or velocity using p = mv. Supports all common unit conversions with impulse reference.

💨 Dynamics📐 p = mv🔄 Conservation
Mass (m)
Unit
Velocity (v)
Unit
⚠️ Please enter valid numbers (velocity must be non-zero for mass/velocity modes).

What Is Momentum?

Momentum is a measure of the quantity of motion possessed by an object. Defined as p = mv (mass × velocity), it captures both how heavy an object is and how fast it moves. The SI unit is kg·m/s (equivalently, N·s — newtons times seconds). Momentum is a vector: it has direction as well as magnitude, and opposing momenta can cancel.

The most powerful application of momentum is the law of conservation of momentum: in a closed system with no external forces, the total momentum remains constant. This holds regardless of whether the collision is elastic (kinetic energy preserved, like billiard balls) or inelastic (energy lost to deformation, like a car crash). Before any collision: Σp_initial = Σp_final.

Closely related is impulse (J = FΔt = Δp): the product of force and the time over which it acts equals the change in momentum. This explains why airbags and crumple zones save lives — they extend the collision time (Δt), reducing the peak force even when the total momentum change is identical. A 70 kg person stopping from 30 m/s to 0 experiences the same Δp = 2,100 kg·m/s whether the stop takes 0.05 s (F = 42,000 N — likely fatal) or 0.5 s (F = 4,200 N — survivable).

Momentum is also fundamental to Newton's second law in its most general form: F_net = dp/dt — the net force equals the rate of change of momentum. For constant mass, this reduces to F = ma, but the momentum form applies even when mass changes (rockets expelling fuel, conveyor belts accumulating material).

Formula Reference Table

Solve ForFormulaNotes
Momentum (p)p = m · vUnits: kg·m/s or N·s
Mass (m)m = p / vv must be non-zero
Velocity (v)v = p / mm must be non-zero
Impulse (J)J = F · Δt = ΔpChange in momentum
Conservationm₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'Before = after collision
Newton's 2nd (general)F = Δp / ΔtNet force = rate of change of p

3 Worked Examples

Example 1
Car on Highway — Find Momentum

A 1,500 kg car travels at 100 km/h. What is its momentum?

  • Convert: v = 100 ÷ 3.6 = 27.78 m/s
  • Apply: p = m × v = 1,500 × 27.78 = 41,667 kg·m/s
  • Compare: An 80 kg sprinter at 10 m/s has p = 800 kg·m/s — 52× less
✓ Momentum = 41,667 kg·m/s
Example 2
Airbag — Impulse Reduces Peak Force

A 75 kg driver decelerates from 60 km/h to rest. Compare crash (0.05 s) vs airbag (0.3 s).

  • Δp: 75 × (60/3.6) = 75 × 16.67 = 1,250 kg·m/s
  • Force without airbag: F = Δp/Δt = 1,250/0.05 = 25,000 N (≈ 34× body weight)
  • Force with airbag: F = 1,250/0.3 = 4,167 N (≈ 5.7× body weight — survivable)
✓ Airbag reduces peak force by 6× — same Δp, 6× longer time
Example 3
Collision — Conservation of Momentum

A 2 kg ball at 5 m/s hits a stationary 3 kg ball and sticks to it. Find final velocity.

  • Before: p_total = 2 × 5 + 3 × 0 = 10 kg·m/s
  • After (stick together): p_total = (2 + 3) × v_f = 5v_f
  • Conservation: 5v_f = 10 → v_f = 2 m/s
✓ Final velocity = 2 m/s (inelastic collision)

Real-World Applications

🚗
Crash Safety
Crumple zones, airbags, and seatbelts all work by extending collision time (Δt), reducing peak force while the total impulse (Δp) remains unchanged. Modern crash standards require specific Δt minimums.
🚀
Rocket Propulsion
Rockets eject mass (exhaust) backward at high velocity. By conservation of momentum, the rocket gains forward momentum. The thrust equation F = v_exhaust × (dm/dt) is the rocket-specific form of F = dp/dt.
🎱
Billiards & Ball Sports
Pool is governed entirely by momentum conservation. The angles and speeds of balls after collision can be calculated precisely — professional players intuitively apply these physics for positional play.
⚽
Sports Science
The momentum transferred to a soccer ball during a penalty kick (p_ball_after − 0) equals the impulse delivered by the foot. High-speed video analysis of kicks measures this impulse for technique optimization.
🌍
Planetary Science
The conservation of angular momentum governs orbital mechanics. A planet moves faster when closer to the sun (Kepler's 2nd law) and slower when farther — angular momentum mv_perp × r stays constant.

Common Mistakes to Avoid

⚠️
Confusing momentum with kinetic energy

Both involve mass and velocity, but KE = ½mv² while p = mv. They have different units (J vs kg·m/s) and behave differently: doubling speed doubles momentum but quadruples kinetic energy. Never interchange them.

⚠️
Forgetting direction (sign) in collisions

Momentum is a vector. If object A moves right (+) and object B moves left (−), their total momentum is p_A − p_B. Dropping signs gives wrong answers in collision problems, especially for head-on scenarios.

⚠️
Applying conservation when external forces act

Conservation of momentum requires zero net external force. A ball rolling along the floor is not a closed system (friction is external). Two colliding billiard balls on a frictionless surface are — but include both balls in the system.

⚠️
Mixing up impulse and momentum

Impulse J = FΔt is a change in momentum (Δp), not momentum itself. J = 500 N·s means the object's momentum changed by 500 kg·m/s, not that it now has 500 kg·m/s of momentum — it may have had momentum before.

⚠️
Using wrong mass units

p = mv requires mass in kg. Using grams (÷1000) or pounds (×0.4536) without conversion gives momentum off by factors of 1,000 or 2.2. Always convert to kg before calculating.

Frequently Asked Questions

What is the unit of momentum? ▾
The SI unit of momentum is kg·m/s (kilogram-meters per second). This is equivalent to N·s (newton-seconds), since 1 N = 1 kg·m/s². Both units appear in physics — kg·m/s in dynamics, N·s when discussing impulse. They are dimensionally identical.
Can two objects have equal momentum but different masses? ▾
Absolutely. p = mv means a 2 kg ball at 10 m/s has the same momentum (20 kg·m/s) as a 10 kg ball at 2 m/s. However, their kinetic energies differ: ½ × 2 × 100 = 100 J vs ½ × 10 × 4 = 20 J. Same momentum, very different energy.
What is the difference between elastic and inelastic collisions? ▾
Both conserve momentum. Elastic collisions also conserve kinetic energy (billiard balls, atomic collisions). Inelastic collisions lose kinetic energy to heat, sound, or deformation — most real collisions are inelastic. Perfectly inelastic collisions (objects stick together) lose the maximum kinetic energy while still conserving momentum.
What is angular momentum? ▾
Angular momentum (L = Iω for rotating objects, or L = mvr for orbital motion) is the rotational equivalent of linear momentum. Like linear momentum, it is conserved when no external torque acts. This is why a spinning figure skater speeds up when pulling in their arms (r decreases, ω increases to keep L = mvr constant), and why planets orbit faster when closer to the sun.
How does the impulse-momentum theorem apply in sports? ▾
Every impact in sport — a tennis serve, cricket drive, golf swing — involves the impulse-momentum theorem: J = FΔt = Δp. A longer contact time (follow-through) allows more impulse for a given force, increasing the ball's momentum change. This is why coaches teach players to follow through: it extends contact time and increases the final velocity of the ball.
Why does a gun recoil when fired? ▾
Conservation of momentum. Before firing, total momentum = 0 (gun and bullet both at rest). After firing, the bullet gains momentum p_bullet = m_bullet × v_bullet forward. For total momentum to remain zero, the gun gains equal backward momentum: m_gun × v_recoil = m_bullet × v_bullet. Hence v_recoil = (m_bullet × v_bullet) / m_gun — much smaller due to the large mass ratio.
Is momentum conserved in everyday situations? ▾
Momentum is always conserved in a closed system. In everyday life, external forces (friction, gravity, normal forces from the Earth) continuously change the momentum of individual objects. When you walk forward, you push back on the Earth — the Earth moves imperceptibly forward. For two-car collisions on a frictionless road, momentum is conserved between the two cars alone. On real roads, friction with the surface leaks momentum to Earth.
How does momentum relate to Newton's Second Law? ▾
Newton's second law in its most general form is F = dp/dt (the net force equals the rate of change of momentum). For constant mass, dp/dt = m × dv/dt = ma, reducing to the familiar F = ma. The momentum form is more general — it correctly handles rockets losing mass, relativistic particles, or any system where mass changes. In fact, Newton originally wrote his second law in terms of motion (momentum), not acceleration.

Related Physics Calculators

💥 Impulse ⚡ Kinetic Energy 🍏 Newton's 2nd Law 🎱 Elastic Collision 💢 Inelastic Collision

Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

Newton's Law of Cooling Calculator →Universal Gravitation Calculator →Newton's Second Law Calculator →Physics Formula Explorer →