When a force acts on an object for a short time — a kick, a catch, a collision — it produces an impulse, and that impulse equals the object’s change in momentum. The same impulse can come from a big force acting briefly or a small force acting for longer, which is the secret behind seatbelts, crumple zones and a good cricket catch.
📘 What you need to know
Impulse is force × time: J = FΔt, measured in newton seconds (N s).
Impulse equals the change in momentum: J = Δp = mv − mu.
These come from Newton’s second law in the form F = Δp / Δt.
Impulse is a vector, in the direction of the resultant force; mind the signs.
Spreading an impact over more time reduces the force for the same change in momentum.
What impulse is
An impulse is what you get when a resultant force acts for a short interval of time:
ImpulseJ = FΔt
For very short impacts it’s hard to measure the force and the contact time directly, so we measure impulse indirectly through the change in momentum it causes. Starting from Newton’s second law, F = Δp / Δt, rearranging gives Δp = FΔt — so the impulse is exactly the change in momentum:
Impulse = change in momentumJ = Δp = mv − mu
where u is the initial velocity and v the final velocity. (These apply when the force F is constant.) Following the units confirms it: since 1 N = 1 kg m s⁻², an impulse in N s is the same as a momentum in kg m s⁻¹.
Impulse as the area under a force–time graph
Because impulse is force × time, it is the area under a force–time graph. For a constant force that’s just a rectangle, F × Δt.
The impulse is the area under the force–time graph — here simply F × Δt for a constant force.
Why a longer impact means a smaller force
Rearranging F = Δp / Δt shows that for a fixed change in momentum, a longer contact time means a smaller force. That’s exactly what a cricket fielder does when catching a fast ball — they draw their hands back to stretch out the impact time and reduce the force on their hands. The same idea protects you with airbags and crumple zones.
Both impacts have the same area — the same impulse and change in momentum — but the longer, softer one needs a far smaller peak force.
Quick reference:J = FΔt = Δp = mv − mu • impulse = area under an F–t graph • 1 N s = 1 kg m s⁻¹ • longer time → smaller force.
Worked examples
WE 1
A tennis ball struck back
A 58 g tennis ball moving left at 30 m s⁻¹ is hit by a racket and returns to the right at 20 m s⁻¹. (a) Find the impulse on the ball. (b) State its direction.
Take the initial motion (left) as positiveu = +30, v = −20, m = 0.058 kg(a) J = m(v − u)= 0.058 × (−20 − 30) = 0.058 × (−50)J = −2.9 N s(b) Negative → opposite to the initial motionimpulse is to the right
WE 2
From force and time to speed
A constant 150 N force acts on a 5.0 kg trolley, initially at rest, for 0.40 s. Find the impulse on the trolley and its final speed.
Impulse: J = F Δt= 150 × 0.40 = 60 N sThis equals the change in momentum: Δp = mv − mu60 = 5.0 × v − 0v = 12 m s⁻¹
💡 Top tips
Impulse = change in momentum. If you can find Δp, you have the impulse, and vice versa.
Watch the signs. When an object rebounds, its final velocity is negative — that’s where the marks are.
Use the area of a force–time graph when the force isn’t constant.
To cut the force, extend the time — same impulse, gentler impact.
N s and kg m s⁻¹ are the same unit, so impulse and momentum can be compared directly.
⚠ Common mistakes
Forgetting the direction change. A bounce flips the sign of the velocity, so Δp is larger than you’d expect.
Using v + u instead of v − u for the change in momentum.
Leaving mass in grams — convert to kg first.
Thinking a softer landing changes the impulse. The impulse (Δp) is the same; only the force and time change.
Impulse and the force–momentum link are two views of the same equation. This page worked with J = Δp; the next one rearranges it to F = Δp / Δt to find the average force in an impact. Up next: Force & Momentum.
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