"What is impulse in physics" is asked by students who have met momentum and now find a second quantity with the same unit, and by anyone who has heard that airbags work "because of impulse" and wants the actual reason. Impulse is simple: a force times the time it acts, which equals the change in momentum it causes. Everything else, from airbags to follow-through, is that one equation rearranged. This page has the formula, where it comes from, the units, the graph, the safety and sport examples with the reasoning spelled out, eight worked examples and the difference between impulse and momentum. The printable has the formulas, the graph and practice on two pages.
Download the impulse and momentum sheet (PDF) (the formulas, the theorem and its derivation, the units, the force-time graph, the safety and sport table, eight worked examples and ten practice questions with answers; two pages, free).
The definition and the formula
Impulse = force × time: J = F Δt
Impulse is the product of a force and the time interval over which it acts. A force of 100 N acting for 0.05 s is an impulse of 5 newton-seconds; so is a force of 5 N acting for 1 s. Impulse is a vector, in the direction of the force.
The impulse-momentum theorem: J = Δp = mv − mu
The impulse delivered to an object equals the change in its momentum. Momentum is p = mv, so the change is m(v − u) for a fixed mass. This is where the formula earns its keep: a collision or a kick is over in milliseconds and the force is impossible to measure directly, but the change in momentum is easy to measure, and from it the impulse and the average force follow.
Where it comes from. Newton's second law in Newton's own form is F = Δp ÷ Δt: force is the rate of change of momentum. Multiply both sides by Δt and FΔt = Δp. (For constant mass, Δp = mΔv and F = ma, the familiar version.)
The units
| Quantity | Formula | Unit | Same as |
|---|---|---|---|
| Impulse | FΔt | newton-second, N·s | kg·m/s |
| Momentum | mv | kg·m/s | N·s |
| Force | Δp ÷ Δt | newton, N | kg·m/s² |
N·s and kg·m/s are the same unit (a newton is a kg·m/s²; multiply by seconds), which is the point of the theorem: impulse and change of momentum are the same thing measured two ways.
Impulse on a force-time graph
A real blow is not a constant force: it rises from zero, peaks and falls back in a few milliseconds. On a graph of force against time, the impulse is the area under the curve. For a constant force the area is a rectangle, F × Δt; for a triangle-shaped blow it is ½ × peak force × duration; for any shape, count the squares or integrate. The average force is the impulse divided by the duration: the height of the rectangle with the same area.
A tennis racket hitting a ball: peak force about 1,000 N, contact about 5 ms, roughly triangular, so impulse ≈ ½ × 1,000 × 0.005 = 2.5 N·s; the 58 g ball's speed changes by 2.5 ÷ 0.058 = 43 m/s.
Why a longer time means a smaller force
In a crash, a catch or a landing, the change in momentum is fixed: the object goes from its speed to zero whatever happens. So the impulse is fixed, and
F = Δp ÷ Δt
says the only thing that can change the force is the time. Make the stop take longer and the force drops in proportion. Everything in the table does exactly that.
| The design | What it lengthens | The result |
|---|---|---|
| Airbag | the time the head and chest take to stop | force on the body down from fatal to survivable |
| Crumple zone | the time the car takes to stop (the front folds over about 0.1 s instead of 0.01 s) | force on the occupants down by about ten times |
| Seat belt (stretches slightly) | the stopping time of the body; also spreads the force over the chest and hips | |
| Bending the knees on landing | the time the body takes to stop | force on the legs and spine down several times |
| Moving the hands back when catching | the time the ball takes to stop | the sting goes |
| Boxing gloves, helmets, padded floors, gym mats | the stopping time of the fist, head or body | |
| Bubble wrap, foam packaging | the stopping time of the parcel's contents | |
| A dropped egg on a cushion against a floor | the stopping time | the cushion's force stays under the shell's breaking force |
The opposite is wanted in sport: a hammer, a bat, a karate strike. There the force is what matters and the contact time is short.
Follow-through uses the same equation the other way. A cricketer, golfer or footballer who follows through keeps the bat, club or foot on the ball for a longer Δt, so FΔt is larger, so the ball's change of momentum, and its speed, is larger. The force is not greater; the contact lasts longer.
Worked examples
| The question | The working | The answer |
|---|---|---|
| A 0.45 kg football at rest is kicked to 20 m/s. Impulse? | Δp = 0.45 × (20 − 0) | 9 N·s |
| The kick lasts 0.01 s. Average force? | F = 9 ÷ 0.01 | 900 N |
| A 0.145 kg baseball arrives at 40 m/s and leaves the bat at 50 m/s the other way. Impulse? | take the leaving direction as positive: Δp = 0.145 × (50 − (−40)) = 0.145 × 90 | 13.05 N·s |
| Contact lasts 0.7 ms. Average force? | 13.05 ÷ 0.0007 | 18,600 N (about 2 tonnes) |
| A 1,200 kg car at 20 m/s hits a wall and stops in 0.1 s (crumple zone). Average force? | Δp = 24,000 kg·m/s; F = 24,000 ÷ 0.1 | 240,000 N |
| The same crash with a rigid car stopping in 0.01 s | 24,000 ÷ 0.01 | 2,400,000 N, ten times more |
| A 60 g egg dropped at 4 m/s stops in 0.005 s on a floor, or 0.1 s on a cushion. Forces? | Δp = 0.24 N·s; ÷ 0.005 and ÷ 0.1 | 48 N (breaks) and 2.4 N (survives) |
| A 2 kg ball hits a wall at 6 m/s and bounces back at 4 m/s. Impulse from the wall? | Δp = 2 × (4 − (−6)) = 2 × 10 | 20 N·s, away from the wall |
| A 70 kg person lands at 5 m/s and stops in 0.5 s with bent knees, or 0.05 s with locked legs | Δp = 350; ÷ 0.5 and ÷ 0.05 | 700 N and 7,000 N |
The sign step in the bounce examples is the one most often missed: when the object reverses direction, the velocities have opposite signs and the change is the sum of their sizes, not the difference.
Impulse and momentum: the difference
| Momentum | Impulse | |
|---|---|---|
| What it is | what a moving body has: p = mv | what changes momentum: J = FΔt |
| Belongs to | an object, at an instant | an interaction, over an interval |
| Zero when | the object is at rest | no force acts, or it acts for no time |
| Conserved? | yes, in a closed system (collisions) | not a conserved quantity; it is a transfer |
| Unit | kg·m/s (= N·s) | N·s (= kg·m/s) |
In a collision between two objects the impulses on each are equal and opposite (Newton's third law, over the same time), which is why total momentum is conserved.
Where it sits in school
Impulse is grade 11 and 12 physics and AP Physics 1 (unit 4, Momentum, where the force-time graph and the theorem are tested) in the United States; GCSE Physics (higher tier: F = Δp ÷ Δt and the safety-feature explanations) and A level in England; Year 11 and 12 Physics in Australia; and CBSE Class 11 (Laws of Motion, where impulse is defined from Newton's second law). The marks go on the sign in bounces, on using the theorem rather than guessing a force, and on writing the safety explanation as "same Δp, longer Δt, smaller F" rather than "it absorbs the impact".
Practice: ten questions
- A force of 40 N acts on a trolley for 3 s. What impulse does it deliver?
- The trolley has mass 8 kg and started from rest. Final speed?
- A 0.06 kg tennis ball is served from rest at 50 m/s. Impulse, and the average force if contact lasts 4 ms?
- A 0.16 kg hockey puck moving at 10 m/s is stopped by a stick in 0.02 s. Average force?
- A 0.2 kg ball hits a wall at 8 m/s and rebounds at 8 m/s. Impulse from the wall?
- The same ball rebounds at only 5 m/s. Impulse?
- A 1,500 kg car stops from 25 m/s in 2 s of braking. Average braking force?
- A force-time graph is a triangle of peak 600 N and base 0.04 s. Impulse?
- A 75 kg skydiver lands at 6 m/s. Compare the force with a 0.6 s roll and a 0.06 s stiff landing.
- Explain in one sentence, using F = Δp ÷ Δt, why a cricketer pulls the hands back when catching.
Answers: 1) 120 N·s. 2) 120 ÷ 8 = 15 m/s. 3) 3 N·s; 3 ÷ 0.004 = 750 N. 4) Δp = 1.6; F = 80 N. 5) 0.2 × (8 − (−8)) = 3.2 N·s. 6) 0.2 × (5 − (−8)) = 2.6 N·s. 7) 37,500 ÷ 2 = 18,750 N. 8) ½ × 600 × 0.04 = 12 N·s. 9) Δp = 450; 750 N against 7,500 N. 10) The ball's change in momentum is fixed, so lengthening the stopping time Δt lowers the force on the hands.
Download the impulse and momentum sheet (PDF)
Where to get help
A 1-on-1 online physics class with a tutor who holds a master's degree in science, for grade 11 and 12 physics, AP Physics 1, GCSE and A level, the Australian years and CBSE Class 11: momentum and impulse worked on the shared screen with the signs written on every velocity, the force-time graph drawn for every blow, and the safety explanations practised in the words the mark scheme wants. The recording and notes after every class, the same tutor every week, US$10 to $13 an hour. The physics page has the courses; the free 30-minute first class is a real lesson, usually on the homework the student brings.
Questions parents ask
1What is impulse in physics?
2What is the impulse formula?
3What is the difference between impulse and momentum?
4Why do airbags reduce injury?
5Why does follow-through matter in sport?
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