Momentum Calculator p = mv: Impulse, Conservation of Momentum and Collisions

About this calculator

Linear momentum is mass multiplied by velocity:

p=mv.

Enter mass and velocity to calculate momentum, or provide momentum and one of the other variables to solve for the missing quantity. Unlike kinetic energy, momentum carries the sign of velocity in one dimension, so direction matters.

Momentum is especially useful for impulse and collision problems because total momentum is conserved in an isolated system even when kinetic energy is not.

Formula and variables

p=mv,\qquad v=\frac{p}{m},\qquad m=\frac{p}{v}

Impulse equals change in momentum:

J=\Delta p=\int F\,dt\approx F_{\text{avg}}\Delta t.

Symbol	Meaning	SI unit	Notes
$m$	mass	kg	positive
$v$	velocity	m/s	signed in one dimension
$p$	momentum	kg·m/s	same sign as velocity
$J$	impulse	N·s	numerically equal to kg·m/s

How to use this calculator

Enter mass and velocity to get momentum.
Use a negative velocity for motion in the chosen negative direction.
To solve for velocity, clear velocity and enter mass plus momentum.
To solve for mass, clear mass and enter velocity plus momentum.
For collision work, calculate each object's signed momentum and add the values algebraically.

Worked examples

Example 1: 1,500 kg car at 20 m/s

At −20 m/s, the momentum is −30,000 kg·m/s while the kinetic energy is unchanged.

Example 2: 0.45 kg ball at 30 m/s

If the ball starts from rest and reaches 30 m/s in 0.01 s, then $\Delta p=13.5$ kg·m/s and the average force is approximately $13.5/0.01=1350$ N.

Example 3: perfectly inelastic collision

Suppose 2 kg moving at 6 m/s hits 4 kg at rest and they stick together. Initial total momentum is 12 kg·m/s, so

Momentum is conserved, but kinetic energy decreases because the collision is inelastic.

Flip velocity from +20 to −20. Momentum changes sign, illustrating the difference between vector-like signed momentum and non-negative kinetic energy.

math

p=(1500)(20)=30000\,kg\cdot m/s.

math

p=(0.45)(30)=13.5\,kg\cdot m/s.

math

(2+4)v_f=12\quad\Rightarrow\quad v_f=2\,m/s.

Principle and derivation

Newton's second law can be written in momentum form as

F=\frac{dp}{dt}.

Integrating over time gives impulse:

\int F\,dt=\Delta p.

For a closed system, internal forces occur in equal-and-opposite pairs, so their impulses cancel in the total. That is why total momentum remains constant during an isolated collision.

Assumptions

The calculation is one-dimensional.
Mass is constant for the basic $p=mv$ form.
Velocity sign represents direction along the chosen axis.
Collision conservation requires negligible external impulse over the collision interval.

Limitations

Supported:: solve any one of $p$ , $m$ , and $v$ from the other two, including signed values.
Approximate only:: real collision contact time and restitution must be measured; average force is only a summary of a time-varying interaction.
Not covered:: full collision solving, multidimensional vectors, angular momentum, variable-mass thrust, relativity, or photon momentum.
No unit conversion:: convert units before entry; impulse, force, and contact time remain separate calculations using $F=\Delta p/\Delta t$ when appropriate.
Numerics:: solving velocity with $m=0$ can produce infinity; extreme values may use scientific notation.

Common pitfalls

Using speed only and losing the sign of direction.
Confusing momentum with kinetic energy: momentum scales with $v$ , energy with $v^2$ .
Assuming kinetic energy is conserved in every collision; it is not conserved in inelastic collisions.
Mixing kg·m/s and other units without conversion.

Typical use cases

Transport safety

Compare pre- and post-impact momentum, reason about crumple zones and airbags, or compare vehicles with different masses and speeds.

Sports

Study momentum transfer in striking and kicking, average impact force with a known contact time, or recoil.

Aerospace and machinery

Build intuition for reaction propulsion, hammers, pile drivers, and mechanical impact loads.

Teaching

Introduce momentum conservation, elastic versus inelastic collisions, and the relationship between momentum and Newton's second law.

FAQ

Can momentum be negative?

Yes. In one dimension, a negative value means the object moves opposite the chosen positive direction.

Is N·s the same as kg·m/s?

Yes. From $F=ma$ , $1\,N\cdot s=1\,kg\cdot m/s$ .

Why do airbags reduce force?

For the same change in momentum, increasing the stopping time reduces average force because $F_{avg}=\Delta p/\Delta t$ .

Privacy

All calculations stay in your browser. This static page does not record user input.

Sources & review

Reviewed by: CalcX 编辑组
Updated: 2026-09-09

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