Multiply a force by the time it acts and you get impulse, so the dimensional formula of impulse is [M¹ L¹ T⁻¹], short form [MLT⁻¹]. That is exactly the bracket of linear momentum.
The match is the impulse momentum theorem in disguise: whatever impulse a body receives shows up as an equal change in its momentum. A bat striking a ball, a seat belt slowing a passenger and an airbag cushioning a head all deliver an impulse, and in every case the answer has units of N s. This page derives the bracket, connects the units, checks the theorem and covers the usual errors. Number problems are worked with the impulse formula.
Quick facts
Dimensional formula
[M¹ L¹ T⁻¹], short form [MLT⁻¹]Defining formula
SI unit
N s = kg m/sVector or scalar
VectorSame dimensions as
Linear momentum
Two routes to the dimensional formula of impulse
From force and time
- Impulse of a constant force: J = F Δt.
- Force has the bracket [MLT⁻²]; the time interval has [T].
- Multiply: one power of time cancels, which leaves [MLT⁻¹], or [M¹ L¹ T⁻¹] written out.
From change in momentum
- J = Δp = m(v − u).
- Mass [M] times a velocity difference [LT⁻¹].
- Result: [MLT⁻¹] again.
For a force that changes during the contact, impulse is the integral ∫F dt, the area under a force time graph. An area on that graph is a force times a time, so it too carries [MLT⁻¹]. The force step is shown on the force page.
Newton seconds and kg m/s
Read [MLT⁻¹] letter by letter and you get kg m s⁻¹. Thinking of it as force times time instead gives the newton second, N s. The two are identical.
Check it: a newton is kg m s⁻²; multiply by a second and one s⁻¹ disappears.
N s is the form most books choose for impulse because it reminds you of force and time. kg m/s is the form most choose for momentum. Either is correct for either quantity.
A cricket bat pushes on the ball with 800 N on average for 0.010 s. Find the impulse in SI and CGS units.
J = F Δt = 800 × 0.010 = 8.0 N s.
For CGS, use the M¹ L¹ powers: 1 N s = 10³ g × 10² cm per s = 10⁵ g cm/s, which is 10⁵ dyne s.
So the impulse is 8.0 × 10⁵ dyne s.
Impulse, momentum and their neighbors
| Quantity | Formula | Comment |
|---|---|---|
| Linear momentum | p = mv | Impulse equals its change |
| Change in momentum | Δp = mv − mu | Impulse momentum theorem |
| Area under an F against t graph | ∫F dt | Same as impulse |
Because impulse and momentum share one bracket, any equation that sets them equal passes a dimensional check. Comparing with nearby brackets helps too: force is [MLT⁻²], one extra T⁻¹, and angular impulse, torque times time, is [ML²T⁻¹], one extra L. See the momentum dimensional formula for p on its own.
Testing the impulse momentum theorem
Check: is the relation F = m(v − u)/t consistent?
Right side: [M] × [LT⁻¹] ÷ [T], which gives [MLT⁻²].
Left side: force, [MLT⁻²]. The equation passes, and multiplying both sides by t returns J = Δp, so the theorem passes as well.
Errors on impulse questions
- Writing [MLT⁻²]. That is the force alone. Multiplying by time removes one T⁻¹.
- Writing [MLT]. The force already has T⁻², so the product cannot have a positive time power.
- Thinking N s and kg m/s are different units. Both collapse to kg m s⁻¹.
- Ignoring the sign of the rebound velocity. The dimensions are unaffected, but the size of the impulse is wrong.
Mechanics brackets for force, momentum and energy are collected on the dimensional formulas page.
Frequently asked questions
Why do impulse and momentum have the same dimensions?
Impulse equals the change in momentum, and a change in a quantity has the same dimensions as the quantity. Both are [MLT⁻¹].
Which unit should an impulse answer carry?
Write N s; kg m/s is an equally valid spelling of the same unit.
Does impulse have a direction?
It does. Impulse is a vector, aimed the same way as the mean force applied while the bodies touch.
How does impulse differ from force in dimensions?
Impulse is force multiplied by time. Force is [MLT⁻²]; impulse is [MLT⁻¹].
What does the area under a force time graph give?
It gives the impulse, with the dimensions [MLT⁻¹] and the unit N s.