Momentum Calculator
Calculate linear momentum (p=mv), impulse, and collision outcomes. Find momentum, mass, or velocity with conservation of momentum principles.
Momentum
50.0000 kg·m/s
Calculation
p = 10 kg × 5 m/s = 50.0000 kg·m/s
Kinetic Energy
125.0000 J
Related Calculators
About This Calculator
Momentum is a measure of an object's mass in motion. It's a fundamental concept in physics that helps explain collisions, explosions, and how forces affect motion over time. This calculator helps you compute momentum, analyze collisions, and understand the relationship between force, time, and momentum change.
What is Momentum? Linear momentum (p) equals mass times velocity: p = mv. It's a vector quantity, meaning it has both magnitude and direction. Momentum is conserved in all collisions when no external forces act on the system.
Key Concepts:
- Momentum: p = mv (mass × velocity)
- Impulse: J = FΔt = Δp (force × time = momentum change)
- Conservation: Total momentum before = total momentum after (in closed systems)
Why Momentum Matters:
- Predicting collision outcomes
- Designing safety systems (airbags, crumple zones)
- Understanding rocket propulsion
- Analyzing sports impacts
- Traffic accident reconstruction
This calculator handles basic momentum, impulse, and collision analysis. For related calculations, see our Kinetic Energy Calculator and Force Calculator.
How to Use the Momentum Calculator
- 1Select your calculation type from the dropdown.
- 2For basic momentum: Enter mass and velocity.
- 3For finding mass or velocity: Enter momentum and the other variable.
- 4For impulse: Enter force and time duration.
- 5For collisions: Enter both objects' masses and initial velocities.
- 6Use negative velocities for opposite directions.
- 7Choose elastic or inelastic collision type.
- 8Review calculated results including energy analysis.
- 9Check momentum conservation (before = after).
- 10Examine energy loss in inelastic collisions.
The Momentum Formula
Understanding the basic relationship.
Linear Momentum
p = m × v
Where:
- p = momentum (kg·m/s)
- m = mass (kg)
- v = velocity (m/s)
Vector Nature
Momentum has direction (same as velocity):
- Moving right: positive momentum
- Moving left: negative momentum
Example Calculations
Car: m = 1500 kg, v = 20 m/s p = 1500 × 20 = 30,000 kg·m/s
Bullet: m = 0.01 kg, v = 500 m/s p = 0.01 × 500 = 5 kg·m/s
Train: m = 100,000 kg, v = 30 m/s p = 100,000 × 30 = 3,000,000 kg·m/s
Comparing Momentum
A slow heavy object can have the same momentum as a fast light object:
- 1000 kg at 10 m/s: p = 10,000 kg·m/s
- 100 kg at 100 m/s: p = 10,000 kg·m/s
Impulse and Momentum Change
How force changes momentum over time.
Impulse-Momentum Theorem
J = F × Δt = Δp = m × Δv
Impulse (J) equals the change in momentum.
Why This Matters
Same momentum change can be achieved with:
- Large force, short time
- Small force, long time
Safety Applications
Airbags: Increase stopping time → reduce force Without airbag: stop in 0.01 s With airbag: stop in 0.1 s Force reduced by 10×!
Example:
Person: m = 70 kg, v = 10 m/s → 0 Δp = 70 × 10 = 700 kg·m/s
Without airbag (t = 0.01 s): F = 700 / 0.01 = 70,000 N (dangerous!)
With airbag (t = 0.1 s): F = 700 / 0.1 = 7,000 N (survivable)
Units
Impulse: N·s or kg·m/s (equivalent) 1 N·s = 1 kg·m/s
Conservation of Momentum
The fundamental principle in collision analysis.
The Law
In a closed system, total momentum is conserved.
p₁ᵢ + p₂ᵢ = p₁f + p₂f
m₁v₁ᵢ + m₂v₂ᵢ = m₁v₁f + m₂v₂f
Conditions
Momentum is conserved when:
- No external forces act on the system
- Or external forces are negligible during collision
Example
Ball 1: m₁ = 2 kg, v₁ᵢ = 5 m/s Ball 2: m₂ = 3 kg, v₂ᵢ = -2 m/s
Initial total: 2(5) + 3(-2) = 10 - 6 = 4 kg·m/s Final total: Must also = 4 kg·m/s
Why It's Conserved
Newton's Third Law: In a collision, forces are equal and opposite. The impulse on object 1 equals the negative impulse on object 2, so total momentum change is zero.
Types of Collisions
Elastic vs. inelastic collisions.
Elastic Collision
- Momentum conserved: m₁v₁ᵢ + m₂v₂ᵢ = m₁v₁f + m₂v₂f
- Kinetic energy conserved: ½m₁v₁ᵢ² + ½m₂v₂ᵢ² = ½m₁v₁f² + ½m₂v₂f²
- Objects bounce apart
- Examples: Billiard balls, atomic collisions
Final velocities (1D elastic): v₁f = [(m₁-m₂)v₁ᵢ + 2m₂v₂ᵢ] / (m₁+m₂) v₂f = [(m₂-m₁)v₂ᵢ + 2m₁v₁ᵢ] / (m₁+m₂)
Inelastic Collision
- Momentum conserved
- Kinetic energy NOT conserved (some lost to heat, sound, deformation)
- Objects may stick together (perfectly inelastic)
- Examples: Car crashes, catching a ball
Perfectly inelastic (objects stick): vf = (m₁v₁ᵢ + m₂v₂ᵢ) / (m₁ + m₂)
Energy Loss Calculation
Energy lost = KE_initial - KE_final
In perfectly inelastic: Max energy loss when objects have equal and opposite momentum.
Special Collision Cases
Important scenarios in collision analysis.
Equal Masses, Elastic
When m₁ = m₂ in elastic collision: Objects exchange velocities!
v₁f = v₂ᵢ v₂f = v₁ᵢ
Example: Newton's cradle
Stationary Target, Elastic
Object 2 initially at rest (v₂ᵢ = 0):
v₁f = v₁ᵢ(m₁-m₂)/(m₁+m₂) v₂f = v₁ᵢ(2m₁)/(m₁+m₂)
Special cases:
- m₁ = m₂: v₁f = 0, v₂f = v₁ᵢ (complete transfer)
- m₁ >> m₂: v₁f ≈ v₁ᵢ, v₂f ≈ 2v₁ᵢ (small object rebounds fast)
- m₁ << m₂: v₁f ≈ -v₁ᵢ, v₂f ≈ 0 (bounces back)
Head-On Collision
When objects approach each other (opposite signs): Higher relative velocity → more dramatic collision
Explosions
Like a reverse perfectly inelastic collision:
- One object becomes two
- Momentum still conserved
- KE increases (from chemical/stored energy)
Applications and Examples
Real-world momentum problems.
Vehicle Collisions
Traffic accident analysis: Two cars collide and stick together. Car 1: 1500 kg at 15 m/s east Car 2: 1000 kg at 20 m/s west
p_total = 1500(15) + 1000(-20) = 22,500 - 20,000 = 2,500 kg·m/s east
vf = 2,500 / 2,500 = 1 m/s east
Sports
Baseball bat hitting ball: Ball: m = 0.145 kg Initial: v = -40 m/s (toward batter) Final: v = +50 m/s (away from batter) Impulse = 0.145 × (50 - (-40)) = 13.05 N·s
Contact time ≈ 0.001 s Average force = 13.05 / 0.001 = 13,050 N
Rocket Propulsion
Rockets work by conservation of momentum:
- Exhaust goes backward (negative p)
- Rocket goes forward (positive p)
- Total momentum stays zero (if started at rest)
Thrust = ṁ × v_exhaust
Pool/Billiards
Understanding momentum transfer helps predict:
- Which direction balls go
- Speed after collision
- When balls stop (energy loss to felt)
Pro Tips
- 💡Momentum = mass × velocity (p = mv).
- 💡Momentum is always conserved in collisions (no external forces).
- 💡Kinetic energy is only conserved in elastic collisions.
- 💡Impulse = force × time = change in momentum.
- 💡Use negative velocities for opposite directions.
- 💡Same impulse: large force short time OR small force long time.
- 💡Equal masses in elastic collision: objects swap velocities.
- 💡Perfectly inelastic: objects stick together, maximum energy loss.
- 💡Momentum has direction—it's a vector quantity.
- 💡Rockets use momentum conservation—no surface needed to push against.
- 💡Airbags work by increasing impact time, reducing force.
- 💡Check your work: total momentum before = total momentum after.
Frequently Asked Questions
Momentum (p = mv) is a vector proportional to velocity. Kinetic energy (KE = ½mv²) is a scalar proportional to velocity squared. In collisions, momentum is always conserved, but KE is only conserved in elastic collisions. A car with twice the speed has twice the momentum but four times the KE.
