Result

Impulse-momentum theorem

J = \int F d t = Δ p

Statement

The impulse delivered by a force, integrated over the time it acts, equals the resulting change in momentum.

Symbols and units

$J$	impulse	N·s
$Δ p$	change in momentum	kg·m/s

When it applies

Any force, constant or varying, applied over a definite time interval.

Don't use when

Requires a well-defined time interval over which the force acts — for an idealized instantaneous collision, use conservation of momentum directly instead of trying to define the force and interval separately.

Derivation

Start from Newton's second law in its momentum form:

F = \frac{d p}{d t}

Integrate both sides over the time the force acts:

\int F d t = \int d p = Δ p

Worked examples

Average force from a bat

A 0.15 kg ball reverses from +20 m/s to −25 m/s in 2.0 ms:

F_{avg} = \frac{Δ p}{Δ t} = \frac{( 0.15 ) ( - 25 - 20 )}{0.0020} \approx - 3400 N

Thousands of newtons, but only for two milliseconds — impulse over a short time can still be modest.

Watch out for

A large force acting briefly can deliver the same impulse as a small force acting for a long time — impulse depends on the product, not the force alone.

Connects to

Newton's second law