Momentum Calculator
Momentum p = mv from mass and velocity in any units, plus 1-D elastic and perfectly inelastic collisions with final velocities and kinetic energy lost.
Momentum Calculator: with the default inputs, momentum of object 1 (kg·m/s) is 6.
Sign gives direction: positive to the right, negative to the left.
- Kinetic energy of object 1
- 9 J
- Total momentum of the system (kg·m/s)
- 6Conserved through any collision.
- Velocity of object 1 after collision
- 3 m/s
- Velocity of object 2 after collision
- —
- Kinetic energy lost in the collision
- 0 J
- Fraction of kinetic energy lost
- 0%
Assumptions
- One-dimensional (head-on) motion; positive is to the right.
- Isolated system during the collision — no external forces or friction.
- Elastic means kinetic energy fully conserved; inelastic means the objects move together afterwards.
How this is worked out
The formula
p = m × v (momentum, kg·m/s) m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂' (conservation of momentum) Perfectly inelastic (stick together): v' = (m₁v₁ + m₂v₂) ÷ (m₁ + m₂) Elastic (kinetic energy also conserved): v₁' = [(m₁ − m₂)v₁ + 2m₂v₂] ÷ (m₁ + m₂) v₂' = [(m₂ − m₁)v₂ + 2m₁v₁] ÷ (m₁ + m₂)
Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- Scenario
- Choose one of 3 options.Single object — momentum p = mv · Two objects — elastic collision (1-D) · Two objects — perfectly inelastic collision (stick together)
- Mass of object 1
- A number.in kg · 0 or more · defaults to 2
- Velocity of object 1
- Sign gives direction: positive to the right, negative to the left.in m/s · defaults to 3
- Mass of object 2(under More options)
- A number.in kg · 0 or more · defaults to 1
- Velocity of object 2(under More options)
- 0 for a stationary target; negative if it moves toward object 1.in m/s · defaults to 0
What you get back
- Momentum of object 1 (kg·m/s)main answer
- Kinetic energy of object 1
- Total momentum of the system (kg·m/s)
- Conserved through any collision.
- Velocity of object 1 after collision
- Velocity of object 2 after collision
- Kinetic energy lost in the collision
- Fraction of kinetic energy lost
What this assumes
- One-dimensional (head-on) motion; positive is to the right.
- Isolated system during the collision — no external forces or friction.
- Elastic means kinetic energy fully conserved; inelastic means the objects move together afterwards.
About this calculator
Momentum is mass times velocity, and its claim to fame is that the total momentum of a closed system never changes — no matter how messy the collision. That single rule lets you predict what happens when a cue ball hits an object ball, a car rear-ends another, or a rocket throws exhaust backward. Enter a mass and velocity for the momentum of one object; open More options and switch the scenario to work out a head-on collision with a second object.
Reading the results
- Momentum is a vector: the sign follows the velocity. Two objects moving toward each other have momenta of opposite sign that partly cancel.
- Total momentum is the same before and after any collision, which is the check the steps show.
- Elastic collisions also conserve kinetic energy (billiard balls, air-track gliders with spring bumpers, gas molecules). Perfectly inelastic collisions are the other extreme: the objects stick, and the maximum possible kinetic energy is lost to heat and deformation (a bullet lodging in a block, coupling railway cars, a tackle).
- Kinetic energy lost is what crumples cars. In a perfectly inelastic collision the fraction lost is m₂ ÷ (m₁ + m₂) when the target is at rest.
Intuition worth keeping
Equal masses colliding elastically simply swap velocities — the cue ball stops dead. A light object hitting a heavy stationary one bounces back at almost the same speed; a heavy one hitting a light one barely slows. Momentum, not kinetic energy, decides recoil: a rifle and its bullet get equal and opposite momentum, but the bullet gets nearly all the kinetic energy because energy goes as p²/2m.
Common mistakes
- Dropping the sign. Velocities in opposite directions must carry opposite signs or the totals are wrong.
- Assuming kinetic energy is conserved in every collision. Only momentum always is; kinetic energy is conserved only in the elastic idealization.
- Applying the 1-D formulas to glancing collisions. Off-centre impacts need two components of momentum; this calculator is strictly head-on.
Where it breaks
Momentum conservation is exact, but the formulas here assume an isolated system (no friction during the brief impact), point masses, and speeds far below light. Real collisions between "elastic" and "perfectly inelastic" are described by a coefficient of restitution between 1 and 0.
Frequently asked questions
▸What is the formula for momentum?
p = mv — mass in kilograms times velocity in meters per second, giving kg·m/s (equivalently N·s). It is a vector, so direction matters.
▸What is conserved in a collision?
Total momentum, always, as long as no outside force acts during the impact. Kinetic energy is conserved only in elastic collisions; in inelastic ones some becomes heat, sound and deformation.
▸How do I find the final velocity in an inelastic collision?
Add the momenta and divide by the combined mass: v' = (m₁v₁ + m₂v₂) ÷ (m₁ + m₂). A 2 kg cart at 3 m/s coupling with a stationary 1 kg cart moves off at 6 ÷ 3 = 2 m/s.
▸What happens when equal masses collide elastically?
They exchange velocities. If one was at rest, the moving one stops and the other leaves with the original velocity — the classic Newton's-cradle and billiards result.
▸What is impulse?
The change in momentum, equal to the average force times the time it acts (J = FΔt = Δp). Airbags and crumple zones lengthen Δt so the same momentum change needs a smaller force.
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