Formula of Impulse: J = FΔt = Δp, Unit and Solved Examples

The formula of impulse is $J = F\Delta t$ : impulse equals the average force multiplied by the time it acts. A bat that pushes on a ball with an average 1200 N for 5.0 milliseconds delivers an impulse of 6.0 N s.

By Newton's second law, that same impulse equals the change in the ball's momentum, $J = \Delta p = m(v - u)$ . This impulse momentum theorem is the main reason impulse appears in the laws of motion chapter. Some books use I instead of J for impulse; the meaning is the same.

Quick facts

Formula
$J = F\,\Delta t$
Impulse momentum theorem
$J = \Delta p = m(v - u)$
SI unit
newton second (N s), same as kg m/s
Dimensional formula
[M¹ L¹ T⁻¹]
Type
Vector, along the average force
From a graph
Area under the force against time curve

Impulse formula physics: symbols and meaning

Symbols in J = FΔt
Symbol	Meaning	SI unit
$J$	Impulse given to the body	N s
$F$	Average net force during contact	N
$\Delta t$	Duration of the contact	s

Impulse is most useful for collisions, kicks and hits, where the force is large, short and changes from moment to moment. You rarely know the force at every instant, but you can measure the velocities before and after. The impulse then tells you the average force once you know the contact time.

Impulse momentum theorem

Start from $F = \frac{\Delta p}{\Delta t}$ and clear the fraction by multiplying through by $\Delta t$ :

$F\,\Delta t = \Delta p \quad\Rightarrow\quad J = mv - mu$

So the net force\'s impulse and the momentum change are one and the same number. The change in momentum formula page works the momentum side; this page works the force and time side. For the force itself, see the average force formula.

Worked examples using the formula of impulse

Example 1: force and time given

A hammer drives a nail with a mean force of 1200 N for 0.0050 s. Find the impulse.

$J = F\Delta t = 1200 \times 0.0050 = 6.0\ \text{N s}$

Example 2: a cricket shot, with km/h

A 160 g ball arrives at 108 km/h and is hit straight back at 20 m/s. Contact lasts 4.0 ms. Find the impulse and the average force.

Convert: 108 ÷ 3.6 = 30 m/s; 160 g = 0.160 kg; 4.0 ms = 0.0040 s. Take the outgoing direction as positive, so u = −30 m/s, v = +20 m/s.

$J = m(v - u) = 0.160 \times (20 - (-30)) = 0.160 \times 50 = 8.0\ \text{N s}$

$F = \frac{J}{\Delta t} = \frac{8.0}{0.0040} = 2000\ \text{N}$

Example 3: impulse from a graph

A force on a puck rises steadily from 0 to 400 N and falls back to 0, all within 0.020 s. Find the impulse.

The graph is a triangle: $J = \frac{1}{2} \times 0.020 \times 400 = 4.0\ \text{N s}$ .

What is the formula of impulse telling you about safety

For a fixed change in momentum, $F = \frac{\Delta p}{\Delta t}$ . Make the time longer and the force goes down. Airbags, helmet foam, gymnastic mats and a fielder pulling the hands back while catching all add time. A long jumper landing in sand stops over perhaps 0.2 s instead of a few hundredths of a second on hard ground, cutting the average force on the legs by several times.

A karate chop works the other way: a fast hand stopping in a very short time gives a large force on the board.

Unit and dimensions of impulse

Force in newtons times time in seconds gives N s. Since 1 N = 1 kg m/s², we get N s = kg m/s, the unit of momentum. Dimensionally, [M L T⁻²] × [T] = [M¹ L¹ T⁻¹], and each step is laid out on the impulse dimensions page. It matches the dimensional formula of momentum, as it must.

Slips with impulse

Milliseconds left as ms. 4.0 ms is 0.0040 s.
No sign on the rebound velocity. Reversing direction makes the impulse larger, not smaller.
Peak force used instead of average. J = FΔt uses the average force; for a graph, use the area.
Impulse confused with work. Work is force times distance, in joules. Impulse is force times time, in N s.

Frequently asked questions

What is the formula of impulse?

Impulse is force times time, J = FΔt. It also equals the change in momentum, J = mv − mu.

Which SI unit is used for impulse?

Impulse is measured in newton seconds (N s). Expand the newton and you get kg m/s, the momentum unit.

Is impulse a vector?

It is a vector quantity. Its direction is that of the average force, and so also that of the momentum change.

What is the difference between impulse and momentum?

Momentum, mv, is a property of a moving body at one moment. Impulse is what a force gives over a time and equals the change in that momentum.

What is an impulsive force?

A large force that acts for a very short time, such as a hammer blow or a bat hitting a ball. Its effect is measured by the impulse rather than the force at any instant.