Momentum Calculator

Calculate momentum, mass, or velocity using p = mv. Includes elastic and inelastic collision calculations.

Momentum Calculator

Solve for:

Mass

Velocity

Momentum Formula:

p = m × v

Momentum (kg·m/s) = Mass (kg) × Velocity (m/s)

Momentum

10000.00 kg·m/s

🎯Momentum

10000.00 kg·m/s

⚖️Mass

1000.00 kg

💨Velocity

10.00 m/s

⚡Kinetic Energy

50000 J

Impulse & Force to Stop:

Impulse needed10000.00 N·s

Force (stop in 1s)10000.00 N

Force (stop in 0.1s)100000.00 N

Collision Calculator

Object 1

Mass (kg)

Velocity (m/s)

Momentum: 10000.00 kg·m/s

Object 2

Mass (kg)

Velocity (m/s)

Momentum: -7500.00 kg·m/s

Collision Type:

Total Initial Momentum

2500.00 kg·m/s

Final Velocity

1.00 m/s

Energy Lost

67500 J (98.2%)

Understanding Momentum

Conservation of Momentum

Total momentum is conserved in all collisions when no external forces act. p₁ᵢ + p₂ᵢ = p₁f + p₂f

Impulse-Momentum Theorem

Impulse equals change in momentum: J = Δp = F × Δt This explains why airbags work - extending collision time reduces force.

What is Momentum?

Momentum is the product of an object's mass and velocity. It represents the "quantity of motion" and is a vector quantity pointing in the direction of velocity.

Property	Symbol	Unit	Description
Momentum	p	kg·m/s	Mass times velocity
Mass	m	kg	Amount of matter
Velocity	v	m/s	Speed with direction
Impulse	J	N·s	Change in momentum

Key insight: A slowly moving heavy truck can have the same momentum as a fast-moving motorcycle.

Momentum Formula

p = mv

Where:

p= Momentum (kg·m/s)
m= Mass (kg)
v= Velocity (m/s)

Conservation of Momentum

The Law of Conservation of Momentum states that the total momentum of an isolated system remains constant:

Condition	Mathematical Form	Implication
Isolated system	p_total = constant	No external forces
Before/after collision	Σp_before = Σp_after	Total momentum conserved
Two objects	m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'	Individual momenta can change

Applications: Collisions, explosions, recoil, rocket propulsion all rely on conservation of momentum.

Conservation of Momentum

m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'

Where:

m₁, m₂= Masses of objects
v₁, v₂= Initial velocities
v₁', v₂'= Final velocities

Impulse and Momentum Change

Impulse is the change in momentum, caused by a force acting over time:

Relationship	Formula	Unit
Impulse definition	J = FΔt	N·s
Impulse-momentum theorem	J = Δp = p_f - p_i	kg·m/s
From Newton's 2nd law	F = Δp/Δt = ma	N

Practical implication: To change momentum gently (less force), increase the time. This is why cars have crumple zones and why you bend your knees when landing.

Impulse-Momentum Theorem

J = FΔt = Δp = m(v_f - v_i)

Where:

J= Impulse (N·s or kg·m/s)
F= Average force (N)
Δt= Time interval (s)
Δp= Change in momentum

Types of Collisions

Collisions are classified by what quantities are conserved:

Type	Momentum	Kinetic Energy	Example
Elastic	Conserved	Conserved	Billiard balls (nearly)
Inelastic	Conserved	NOT conserved	Car crash
Perfectly inelastic	Conserved	Maximum loss	Objects stick together
Explosion	Conserved	Increases	Firework bursting

Collision Equations

Elastic: ½m₁v₁² + ½m₂v₂² = ½m₁v₁'² + ½m₂v₂'² Perfectly inelastic: v' = (m₁v₁ + m₂v₂)/(m₁ + m₂)

Where:

v'= Final common velocity (inelastic)
KE= Kinetic energy (½mv²)

Elastic Collision Formulas

For perfectly elastic collisions in one dimension:

Scenario	Formula for v₁'	Formula for v₂'
General case	v₁' = ((m₁-m₂)v₁ + 2m₂v₂)/(m₁+m₂)	v₂' = ((m₂-m₁)v₂ + 2m₁v₁)/(m₁+m₂)
Equal masses	v₁' = v₂	v₂' = v₁
Object 2 at rest	v₁' = (m₁-m₂)v₁/(m₁+m₂)	v₂' = 2m₁v₁/(m₁+m₂)
m₁ >> m₂, v₂=0	v₁' ≈ v₁	v₂' ≈ 2v₁
m₁ << m₂, v₂=0	v₁' ≈ -v₁	v₂' ≈ 0

Real-World Applications

Momentum principles are used in many technologies and safety systems:

Application	Principle Used	How It Works
Airbags	Impulse (FΔt = Δp)	Increase Δt to reduce F
Rocket propulsion	Conservation	Expel mass backward, gain momentum forward
Gun recoil	Conservation	Bullet momentum forward = gun momentum backward
Pool/billiards	Elastic collisions	Transfer momentum and energy
Crash testing	Impulse analysis	Design crumple zones
Spacecraft docking	Inelastic collision	Objects combine, share momentum

Angular Momentum

Angular momentum is the rotational analog of linear momentum:

Quantity	Linear	Angular
Momentum	p = mv	L = Iω
Inertia	Mass (m)	Moment of inertia (I)
Velocity	v	Angular velocity (ω)
Force relation	F = dp/dt	τ = dL/dt
Conservation	If ΣF = 0, p = const	If Στ = 0, L = const

Example: Figure skaters spin faster when pulling arms in (decreasing I, so ω increases to keep L constant).

Angular Momentum

L = Iω = mvr (for point mass)

Where:

L= Angular momentum (kg·m²/s)
I= Moment of inertia (kg·m²)
ω= Angular velocity (rad/s)
r= Radius (m)

Worked Examples

Calculate Momentum

Problem:

A 1,200 kg car travels at 25 m/s. What is its momentum?

Solution Steps:

1Identify values: m = 1,200 kg, v = 25 m/s
2Apply formula: p = mv
3Calculate: p = 1,200 × 25 = 30,000 kg·m/s

Result:

Momentum = 30,000 kg·m/s

Perfectly Inelastic Collision

Problem:

A 2 kg ball moving at 5 m/s collides and sticks to a 3 kg ball at rest. Find the final velocity.

Solution Steps:

1Total initial momentum: p_i = m₁v₁ + m₂v₂ = 2(5) + 3(0) = 10 kg·m/s
2After collision, combined mass: m_total = 2 + 3 = 5 kg
3Conservation: p_f = p_i, so (5)v' = 10
4Final velocity: v' = 10/5 = 2 m/s

Result:

Final velocity = 2 m/s in original direction

Impulse from Force

Problem:

A bat exerts 15,000 N on a 0.145 kg baseball for 0.002 s. What velocity change occurs?

Solution Steps:

1Calculate impulse: J = FΔt = 15,000 × 0.002 = 30 N·s
2Impulse equals momentum change: J = mΔv
3Solve for Δv: Δv = J/m = 30/0.145 = 207 m/s
4This is the change in velocity (could reverse direction)

Result:

Velocity change = 207 m/s (≈ 463 mph)

Tips & Best Practices

✓Choose a positive direction and stick with it throughout the problem
✓Momentum is ALWAYS conserved in isolated systems (no external forces)
✓Kinetic energy is only conserved in perfectly elastic collisions
✓For perfectly inelastic collisions (objects stick), use v' = (m₁v₁ + m₂v₂)/(m₁+m₂)
✓Impulse = Force × Time = Change in momentum (J = FΔt = Δp)
✓To reduce impact force, increase impact time (padding, crumple zones)
✓1 kg·m/s = 1 N·s (units of momentum and impulse are equivalent)

Frequently Asked Questions

Momentum is conserved in isolated systems where no external forces act. In real-world collisions, external forces like friction or gravity may cause momentum loss to the surroundings, but total momentum of the universe is still conserved. For practical problems, we assume the collision happens fast enough that external forces are negligible during the collision.

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. A car doubling its speed doubles its momentum but quadruples its kinetic energy.

From J = FΔt = Δp, if momentum change (Δp) is fixed, then increasing time (Δt) decreases force (F). Airbags and crumple zones extend the collision time from milliseconds to tens of milliseconds, reducing peak forces on occupants by a factor of 10-20, potentially making survivable what would otherwise be fatal.

Rockets don't push against air—they use conservation of momentum. By expelling exhaust gases backward at high velocity, the rocket gains equal momentum forward. This works in vacuum: p_rocket + p_exhaust = constant. The thrust depends on exhaust mass flow rate and exhaust velocity: F = (dm/dt)v_exhaust.

When two objects of equal mass collide elastically, they exchange velocities. If one is at rest, it moves with the first object's initial velocity while the first stops completely (like in Newton's cradle). This is the most efficient transfer of momentum and energy possible. The math works out perfectly when m₁ = m₂.

Yes! Since momentum is a vector (p = mv), its sign indicates direction. If you define rightward as positive, leftward motion has negative momentum. In collision problems, be careful to assign consistent signs based on your chosen coordinate system. Negative momentum doesn't mean less motion—it means opposite direction.

Sources & References

Halliday, Resnick & Walker - Fundamentals of Physics (2024)
MIT OpenCourseWare - Classical Mechanics (2024)
NHTSA - Crash Safety Research (2024)
NASA - Rocket Propulsion (2024)

Last updated: 2026-01-22

Momentum Calculator