Impulse and Momentum Calculator

Calculate impulse, momentum, force, and velocity relationships

Input any eight values to solve for the remaining one using impulse-momentum relationships.

Last updated: December 24, 2025

CreatorFrank Zhao

Impulse and Momentum Variables

Velocity change (Δv)

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m/s

Initial velocity (v₁)

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m/s

Final velocity (v₂)

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m/s

Mass (m)

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Force (F)

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Time interval

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sec

Impulse (J)

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N⋅s

Initial momentum (p₁)

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N⋅s

Final momentum (p₂)

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N⋅s

1Impulse from ForceJ = F × t

J = F \cdot t

2Impulse from VelocityJ = m × Δv

J = m \cdot \Delta v

3Momentump = m × v

p = m \cdot v

4Velocity ChangeΔv = v₂ - v₁

\Delta v = v_2 - v_1

5Impulse TheoremJ = p₂ - p₁

J = p_2 - p_1

6Force from ImpulseF = (m × Δv) / t

F = \frac{m \cdot \Delta v}{t}

7Time from Impulset = J / F

t = \frac{J}{F}

JImpulse

FForce

tTime

mMass

vVelocity

pMomentum

Introduction / Overview

Impulse and momentum are two sides of the same story: momentum describes how hard something is to stop, and impulse describes what it takes to change that motion.

What it helps with

Solve for impulse, momentum, velocity change, force, time interval, or mass by entering values you already know.

Who uses it

Students, coaches, engineers, and curious builders—anyone dealing with impacts, braking, throws, recoil, or short bursts of force.

Pair it with the conservation of momentum calculator for collisions, or the momentum calculator if you want a dedicated momentum-only workflow.

Formula for momentum

Momentum $p$ is a vector that depends on mass $m$ and velocity $v$ . In the simplest one‑direction case, the relationship is:

p = m\cdot v

When momentum changes from $p_1$ to $p_2$ , that change is called impulse $J$ :

J = \Delta p

=

p_2 - p_1

=

m\,v_2 - m\,v_1

=

m\,(v_2 - v_1)

=

m\,\Delta v

The symbol $\Delta$ means “change over an interval”. For example, $\Delta p$ is the difference between final and initial momentum.

Impulse equation

If a force $F$ acts over a time interval $t$ , the impulse is:

J = F\cdot t

Units

In SI, impulse and momentum share the same units: $\mathrm{N\cdot s}$ which is equivalent to $\mathrm{kg\cdot m/s}$ .

Sign matters

A negative $J$ means the impulse is opposite the object’s current direction of motion.

If you’re working with rockets and engines, you may also like our specific impulse calculator (different topic, but the naming trips people up).

How to calculate impulse (three practical routes)

1
From momentum change
Enter $p_1$ and $p_2$ (or mass and both velocities) and use $J = \Delta p = p_2 - p_1$ .
2
From mass and velocity change
If you know the object’s mass and how much its speed changed, use $J = m\,\Delta v$ .
3
From force and time
If the force is roughly constant over the interval, enter $F$ and $t$ and use $J = F\,t$ .

Step-by-step example: stopping a ball

Suppose a ball of mass $m=0.160\ \mathrm{kg}$ is moving at $v_1=2.5\ \mathrm{m/s}$ and you bring it to rest ( $v_2=0$ ). The impulse required is:

J = m\,(v_2 - v_1)

=

0.160\,(0 - 2.5)

=

-0.4\ \mathrm{N\cdot s}

The negative sign tells you the impulse points opposite the motion—exactly what you’d expect when stopping something.

If you’re exploring recoil or impacts, you may also like our recoil energy calculator for a complementary perspective.

Real-world examples / use cases

Braking a cart

A $12\ \mathrm{kg}$ cart slows from $3\ \mathrm{m/s}$ to $0.5\ \mathrm{m/s}$ .

J = m\,(v_2-v_1)

=

12\,(0.5-3)

=

-30\ \mathrm{N\cdot s}

Interpretation: you need $30\ \mathrm{N\cdot s}$ of impulse opposite the motion.

Short impact

A constant force of $400\ \mathrm{N}$ acts for $0.015\ \mathrm{s}$ .

J = F\,t

=

400\cdot 0.015

=

6\ \mathrm{N\cdot s}

That’s the momentum change delivered by the hit.

Finding average force

If you measure impulse $J=18\ \mathrm{N\cdot s}$ over $t=0.12\ \mathrm{s}$ , then

F = \frac{J}{t}

=

\frac{18}{0.12}

=

150\ \mathrm{N}

Useful for “average force” estimates when the force curve is complicated.

If you’re analyzing collisions, follow up with the conservation of momentum calculator to connect before/after velocities to the same momentum ideas.

Common scenarios / when to use

Stopping or starting motion

Use $J=m\,\Delta v$ to estimate the impulse needed for a speed change.

Short force bursts

For impacts where you know a rough average force, try $J=F\,t$ .

Measuring momentum change

If you already have initial and final momentum, go straight to $J=\Delta p$ .

Comparing two designs

Compare how different masses or contact times affect impulse requirements.

Classroom checks

Validate hand calculations quickly, then focus on interpretation.

Collision follow-ups

Use it together with conservation of momentum for before/after states.

Not a great fit if the force varies wildly and you need precise peak forces. In that case, you’d typically integrate a force‑time curve ( $J=\int F\,dt$ ) rather than rely on a single average.

Tips & best practices

Be consistent with direction

Pick a positive direction and stick to it. A negative $J$ is not “wrong”—it often means “slowing down”.

Use the simplest input route

If you already know $p_1$ and $p_2$ , don’t detour—use $J=p_2-p_1$ .

Watch your units

Make sure mass is in $\mathrm{kg}$ if velocity is in $\mathrm{m/s}$ so the result lands in $\mathrm{N\cdot s}$ .

Average force is an approximation

When you use $J=F\,t$ , you’re implicitly using an average $F$ over the interval.

Related concepts / background

Impulse-momentum theorem

The core idea is simply $J=\Delta p$ . If the impulse is opposite motion, momentum decreases.

Why “longer contact time” helps

For the same required impulse $J$ , increasing $t$ reduces the average force since $F=J/t$ .

Want the conservation-angle? Jump to conservation of momentum to see how two-body systems exchange momentum in collisions.

Frequently asked questions (FAQs)

How do I calculate impulse from momentum?

Use the difference between final and initial momentum:

J = \Delta p = p_2 - p_1

What is the impulse-momentum theorem?

It states that the impulse applied to an object equals its change in momentum: $J=\Delta p$ . The sign of $J$ depends on direction.

Are impulse and momentum the same thing?

They’re related, but not identical. Momentum $p$ describes motion at an instant, while impulse $J$ describes a change in momentum over time. The bridge is $J=\Delta p$ .

What impulse is required to stop a ball if m = 160 g and v = 2.5 m/s?

Convert mass to kilograms and apply $J=m\,(v_2-v_1)$ with $v_2=0$ .

J = 0.160\,(0 - 2.5)

=

-0.4\ \mathrm{N\cdot s}

The negative sign indicates the impulse is opposite the ball’s motion.

When should I use J = F·t instead of J = mΔv?

Use $J=F\,t$ when you have a reasonable estimate for the (average) force over a known time interval. Use $J=m\,\Delta v$ when you trust your mass and velocity measurements more.

Limitations / disclaimers

Using $J=F\,t$ assumes a constant or average force over the time interval.
Real collisions can involve rotation, deformation, and varying contact forces that aren’t captured by a single number.
This tool is educational and informational; it’s not a substitute for professional engineering or safety advice.

External references / sources

If you want a more formal derivation and context, these are solid starting points:

The Physics Hypertextbook — Momentum
Clear explanations and standard definitions.
OpenStax — University Physics (Vol. 1)
Free textbook coverage of momentum and impulse concepts.

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