Last updated: 2026-06-14

Momentum and Impulse

Momentum measures how much motion an object carries: $p = m \cdot v$, the product of mass and velocity. Impulse measures the effect of a force acting over time: $J = F \cdot t$. The two are linked by the impulse–momentum theorem, which states that the impulse applied to an object equals its change in momentum: $F \cdot t = m \cdot \Delta v$. Both momentum and impulse are vector quantities and share the same units, kilogram-metres per second (kg·m/s), equivalent to newton-seconds (N·s).

This calculator finds momentum from mass and velocity, impulse from force and time, or the average force from a change in momentum.

The impulse–momentum theorem

Newton's second law is usually written $F = m \cdot a$, but acceleration is just the rate of change of velocity. Rewriting it over a time interval gives $F \cdot t = m \cdot \Delta v$: the impulse delivered equals the momentum gained or lost. This form is far more useful for impacts and collisions, where the force varies rapidly and is hard to measure instant by instant, but the overall change in momentum is easy to determine.

Formulas

Impulse equals the change in momentum, so $J = m \cdot \Delta v$ whenever a mass changes velocity, regardless of how the force varies during the interval.

Worked example

A 0.145 kg baseball is pitched at 40 m/s and struck straight back at 40 m/s in the opposite direction. Taking the original direction as positive, the change in velocity is $\Delta v = -40 - 40 = -80$ m/s, so the impulse is:

If the bat is in contact with the ball for 0.7 ms (0.0007 s), the average force is enormous:

This is why a short contact time produces such a violent force, and why measuring the impulse is more practical than measuring the peak force directly.

Why airbags and crumple zones work

In a crash, an occupant's momentum must drop to zero, so the impulse — the change in momentum — is fixed. Because $J = F \cdot t$ is constant, lengthening the time over which the momentum changes lowers the average force. Airbags, crumple zones, padded dashboards, and even bending your knees on landing all work by stretching the stopping time, which reduces the force the body experiences. Doubling the stopping time roughly halves the peak force.

Conservation of momentum

In any collision or explosion where no external force acts, the total momentum of the system is conserved — the vector sum before equals the vector sum after. This principle, which follows directly from Newton's third law, governs everything from billiard balls and car crashes to rocket propulsion and subatomic particle interactions.