Momentum Calculator
Calculate linear momentum (p = mv) and impulse (J = FΔt) for moving objects. Enter mass and velocity to find momentum, plus optional force and time for impulse calculations. See also our Force Calculator and Kinetic Energy Calculator.
How to Calculate Momentum
Momentum is one of the most important quantities in physics, representing the "quantity of motion" of an object. Defined as the product of mass and velocity (p = mv), momentum is a vector quantity — it has both magnitude and direction. The concept was developed by René Descartes and later refined by Isaac Newton, who expressed his Second Law in terms of momentum: force equals the rate of change of momentum.
To calculate momentum, multiply the mass of the object by its velocity. The SI unit is kilogram-meters per second (kg⋅m/s), which is equivalent to newton-seconds (N⋅s). Unlike kinetic energy, momentum is directly proportional to velocity (not velocity squared), making it a linear measure of motion. This distinction is important in collision analysis where momentum is always conserved but kinetic energy may not be.
Impulse is the change in momentum caused by a force acting over time: J = FΔt = Δp. This relationship explains why airbags and crumple zones save lives — by increasing the time of deceleration, they reduce the force experienced by occupants while producing the same change in momentum. A longer stopping time means a smaller force for the same momentum change.
Momentum and Impulse Formulas
Linear Momentum:
p = mv
p = momentum (kg⋅m/s), m = mass (kg), v = velocity (m/s)
Impulse-Momentum Theorem:
J = FΔt = Δp = m(v₂ - v₁)
Conservation of Momentum:
m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂' (collisions)
Elastic Collision (1D):
v₁' = [(m₁-m₂)v₁ + 2m₂v₂]/(m₁+m₂)
v₂' = [(m₂-m₁)v₂ + 2m₁v₁]/(m₁+m₂)
Relationship to KE:
KE = p²/(2m)
p = √(2m×KE)
Example Calculation
A 5 kg bowling ball moves at 10 m/s. Calculate its momentum and the force needed to stop it in 0.5 seconds:
Given: m = 5 kg, v = 10 m/s
Momentum: p = mv = 5 × 10 = 50 kg⋅m/s
KE = ½mv² = ½ × 5 × 100 = 250 J
Verify: KE = p²/(2m) = 2500/10 = 250 J ✓
To stop in 0.5 seconds (v₂ = 0):
Δp = m(v₂-v₁) = 5(0-10) = -50 kg⋅m/s
F = Δp/Δt = -50/0.5 = -100 N (opposing motion)
To stop in 0.05 seconds (hard wall):
F = -50/0.05 = -1000 N (10× more force!)
Momentum Reference Table
| Mass (kg) | Velocity (m/s) | Momentum (kg⋅m/s) | Object |
|---|---|---|---|
| 0.003 | 900 | 2.7 | Bullet |
| 0.06 | 60 | 3.6 | Tennis ball serve |
| 0.145 | 40 | 5.8 | Baseball pitch |
| 0.43 | 30 | 12.9 | Soccer ball kick |
| 5 | 10 | 50 | Bowling ball |
| 70 | 2 | 140 | Walking person |
| 70 | 10 | 700 | Sprinting person |
| 100 | 5 | 500 | Motorcycle |
| 1000 | 14 | 14000 | Car at 50 km/h |
| 1500 | 28 | 42000 | Car at 100 km/h |
| 40000 | 28 | 1120000 | Truck at 100 km/h |
| 400000 | 300 | 120000000 | Airplane |
Frequently Asked Questions
What is momentum?
Momentum is the product of an object's mass and velocity: p = mv. It is a vector quantity measured in kg⋅m/s (or equivalently, N⋅s). Momentum represents the "quantity of motion" — a heavy slow object can have the same momentum as a light fast object. It is one of the fundamental conserved quantities in physics, meaning total momentum is preserved in all interactions.
What is the difference between momentum and kinetic energy?
Momentum (p = mv) is a vector proportional to velocity, while kinetic energy (KE = ½mv²) is a scalar proportional to velocity squared. In collisions, momentum is always conserved, but kinetic energy is only conserved in elastic collisions. Two objects with the same momentum can have different kinetic energies. They are related by KE = p²/(2m). Momentum determines what happens after a collision; KE determines how much energy is available.
What is impulse and how does it relate to momentum?
Impulse (J) is the change in momentum of an object: J = Δp = FΔt. It equals the force multiplied by the time the force acts. A large force for a short time produces the same impulse as a small force for a long time. This is why catching a ball with "soft hands" (longer contact time) reduces the force on your hands — same momentum change, longer time, less force.
Why is momentum conserved in collisions?
Momentum conservation follows from Newton's Third Law: when two objects interact, they exert equal and opposite forces on each other for the same duration. The impulse on object 1 is equal and opposite to the impulse on object 2, so the total momentum change is zero. This applies to all collisions (elastic and inelastic) as long as no external forces act on the system.
What is an elastic vs inelastic collision?
In an elastic collision, both momentum AND kinetic energy are conserved (e.g., billiard balls, atomic collisions). In an inelastic collision, momentum is conserved but kinetic energy is not — some energy converts to heat, sound, or deformation. A perfectly inelastic collision is when objects stick together (maximum KE loss). Most real collisions are partially inelastic.
How do airbags use the impulse-momentum theorem?
In a car crash, the occupant's momentum must change from mv to zero (Δp is fixed by the crash speed). Since J = FΔt = Δp, increasing the stopping time (Δt) reduces the force (F). An airbag increases the deceleration time from about 5ms (hitting the dashboard) to about 50ms, reducing the force by a factor of 10. Crumple zones work the same way for the vehicle structure.
Conservation of Momentum in Practice
Conservation of momentum is used in rocket propulsion (exhaust momentum backward = rocket momentum forward), ballistic pendulums (measuring bullet speed), particle physics (analyzing collision products), sports (analyzing impacts and rebounds), and accident reconstruction (determining pre-collision speeds from post-collision evidence). It is one of the most powerful problem-solving tools in physics because it applies regardless of the internal forces between objects.
Angular Momentum
Angular momentum (L = Iω or L = r × p) is the rotational analog of linear momentum. Like linear momentum, angular momentum is conserved when no external torque acts on a system. This explains why ice skaters spin faster when they pull their arms in (reducing moment of inertia increases angular velocity to conserve L), why gyroscopes resist tilting, and why planets orbit in stable ellipses. Angular momentum conservation is fundamental to understanding rotating systems from atoms to galaxies.