Impulse-Momentum Calculator: J = m(v2 - v1)
Apply the impulse-momentum theorem: find the impulse, mass or either velocity from J = m(v2 - v1), and the average force over a time. Free, in your browser.
Impulse is the change in momentum: J = m × (v₂ − v₁). Leave one field empty to solve for it.
Fill in all but one field: the empty field is the one you solve for.
About the Impulse-Momentum Calculator
This page opens on the impulse-momentum theorem: the impulse on an object equals its change in momentum, J = m(v₂ − v₁). Fill in all but one of Impulse, Mass, Initial velocity and Final velocity, and the empty one appears as you type. The calculator shows the formula with the solved symbol highlighted and a one-line substitution such as 2 kg × (8 m/s − 3 m/s) = 10 kg·m/s.
The theorem is a direct result of Newton's second law. A net force F acting for a time Δt changes the momentum by F × Δt, so you can find the final speed after a push, or the average force in a collision, without knowing the acceleration at every instant. Use the mode switch to move to Force and time, where J = F × Δt.
How to use the impulse-momentum calculator
Steps
- Enter the mass and the two velocities to find the impulse, or enter the impulse with any two of mass, v₁ and v₂ to find the third.
- Choose units. Velocity offers m/s, km/h, mph, ft/s and knots, mass offers kg, g and lb, and impulse offers N·s, kN·s, kg·m/s and lbf·s.
- Read the answer under the formula, then switch the mode to find the average force over a time.
Mass must be greater than zero. Velocities and impulse may be negative, and negative values are shown in brackets in the substitution step. If the two velocities are equal, the mass cannot be found, and an inline message says so.
Formula
Here v₁ is the initial velocity and v₂ is the final velocity. The change in velocity can be negative, which gives a negative impulse in the direction you chose as positive.
Worked examples
Example 1: find the final velocity
A 2 kg cart starts at 3 m/s and receives an impulse of 10 kg·m/s. Its final velocity is v₂ = 3 + 10 ÷ 2 = 8 m/s.
Example 2: a ball bouncing off a wall
A 0.5 kg ball hits a wall at 10 m/s and rebounds at 8 m/s the other way. Taking the incoming direction as positive, v₁ = 10 and v₂ = −8, so J = 0.5 × (−8 − 10) = −9 kg·m/s. If the contact lasts 0.01 s, the average force is −9 ÷ 0.01 = −900 N.
Example 3: a car stopping
A 1,000 kg car brakes from 20 m/s to rest, so J = 1,000 × (0 − 20) = −20,000 N·s. Over 0.1 s the average force is 200,000 N in magnitude. Over 0.5 s, the longer stopping time of a crumple zone, it falls to 40,000 N.
Why it matters
Many design choices come down to this theorem. Airbags, helmets, padded floors and bent knees when landing all lengthen the time over which momentum is removed. Since the impulse needed is fixed by the change in momentum, a longer time means a smaller average force on the body.
Privacy
Nothing is uploaded; every calculation runs in your browser and never leaves your device.
Related tools
The Impulse Calculator opens on force and time, the Momentum Calculator solves p = m × v, and the Force Calculator applies F = m × a.
Frequently asked questions
What is the impulse-momentum theorem?
It states that the impulse on an object equals its change in momentum: J = m(v₂ − v₁). Impulse is force times the time it acts, so a force over time changes momentum.
How do I find the final velocity?
Use v₂ = v₁ + J ÷ m. A 2 kg cart at 3 m/s that receives 10 kg·m/s of impulse ends up at 3 + 10 ÷ 2 = 8 m/s. Leave the final velocity field empty and fill in the other three.
How do I find the average force?
Divide the impulse by the contact time. Switch the mode to Force and time, enter the impulse and the time, and leave force empty. An impulse of 9 N·s over 0.01 s means 900 N.
Why is the impulse sometimes negative?
The sign shows direction. If an object slows down or reverses, the velocity change is negative in the direction you chose as positive, so the impulse is negative too. A 2 kg object slowing from 8 to 3 m/s has J = −10 kg·m/s.
Why do airbags and crumple zones help?
They make the stopping time longer. The impulse needed to stop you is fixed by your mass and speed, so a longer time means a smaller average force on your body.
Which units are supported?
Velocity offers m/s, km/h, mph, ft/s and knots. Mass offers kg, g and lb. Impulse offers kg·m/s, N·s, kN·s and lbf·s. Each field has its own unit.
Is anything uploaded?
No. The impulse, mass and velocities you enter are calculated locally in your browser, and nothing is uploaded, stored or shared with a server, so any private test data stays on your device.