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Momentum Dimensional Formula

Dimensional formulas of physical quantitiesClass 9Class 11Updated 1 Nov 2026

[p]=[M1L1T−1][p] = [M^{1} L^{1} T^{-1}]
SI unit
kg m/s (= N s)
Dimensions
[M¹ L¹ T⁻¹]
where
  • M dimension of mass kg
  • L dimension of length m
  • T dimension of time s

The momentum dimensional formula is [MLT⁻¹], written in full as [M¹ L¹ T⁻¹]. Linear momentum depends on mass and length to the first power and on time to the power minus one.

It follows from p = mv: a mass in kilograms carried along at a velocity in metres per second. A loaded truck and a fast cricket ball can have the same momentum, and both reduce to the same three base dimensions. This page works the derivation, the unit kg m/s, the link to impulse and two checks. Values and collision problems are on the momentum formula page.

Quick facts

  • Dimensional formula

    [M¹ L¹ T⁻¹], short form [MLT⁻¹]

  • Defining formula

    p=mvp = mv

  • SI unit

    kg m/s, equal to N s

  • CGS unit

    g cm/s; 1 kg m/s = 10⁵ g cm/s

  • Vector or scalar

    Vector, along the velocity

  • Same dimensions as

    Impulse, change in momentum

Momentum dimensional formula step by step

Linear momentum is the product of mass and velocity. Replace each factor by its base quantities.

  1. Defining relation: p=mvp = m v.
  2. Mass needs no breaking down: [m]=[M][m] = [M].
  3. Velocity is displacement over time, [v]=[L][T]=[LT−1][v] = \frac{[L]}{[T]} = [L T^{-1}].
  4. Product: [p]=[M]⋅[LT−1][p] = [M] \cdot [L T^{-1}], so [p]=[MLT−1][p] = [M L T^{-1}].

That is the dimensional formula of linear momentum. Whether a worksheet asks for the dimension formula of linear momentum or the dimensional formula for momentum, this is the bracket it wants. Note that the word "linear" just separates it from angular momentum; the bracket above is what every board means by the momentum of a moving body. For the middle step in full, open the velocity dimensions derivation.

What the bracket says about the unit

Read M as kilogram, L as metre and T⁻¹ as per second. The SI unit is kg m s⁻¹, normally written kg m/s. Momentum has no special named unit.

A second form is the newton second. One newton is kg m s⁻², and multiplying by one second removes one power of s⁻¹, leaving kg m s⁻¹. So 1 N s = 1 kg m/s exactly. Answer keys use either.

In CGS the unit is g cm/s. A kilogram is a thousand grams and a metre is a hundred centimetres, so the factor from kg m/s to g cm/s is 10⁵.

Quantities that share [MLT⁻¹]

Same bracket as momentum: [M¹ L¹ T⁻¹]
QuantityFormulaWhy it matches
ImpulseJ=F ΔtJ = F \, \Delta tForce [MLT⁻²] times time [T]
Change in momentumΔp=mv−mu\Delta p = m v - m uA difference of two momenta
Photon momentump=hλp = \frac{h}{\lambda}[ML²T⁻¹] divided by [L]

The match between impulse and momentum is the impulse momentum theorem written in dimensions: FΔt=ΔpF\Delta t = \Delta p. Any equation that sets one equal to the other is at least consistent. See the dimensional formula of impulse for the same bracket built from force.

Not to be confused with

Students often answer a momentum dimensions question with a nearby bracket. This table puts the close ones together.

Momentum compared with quantities one step away
QuantityDimensionsDifference from momentum
Momentum p[MLT⁻¹]reference
Force F = dp/dt[MLT⁻²]one more T⁻¹
Kinetic energy p²/2m[ML²T⁻²]p squared over a mass
Angular momentum r × p[ML²T⁻¹]one more L

Force is the time rate of momentum, so dividing by T gives the force bracket. Multiplying by a lever arm gives angular momentum.

Worked checks on momentum

Check: is K = p²/(2m) dimensionally correct?

Right side: [MLT−1]2[M]=[M2L2T−2][M]=[ML2T−2]\frac{[M L T^{-1}]^{2}}{[M]} = \frac{[M^{2} L^{2} T^{-2}]}{[M]} = [M L^{2} T^{-2}].

Kinetic energy is an energy, [ML²T⁻²]. The sides agree. The factor 2 is a pure number, so dimensions cannot confirm it.

A ball of 160 g moves at 36 km/h. Find its momentum in SI units.

Convert first: m = 0.160 kg, v = 36 × 1000/3600 = 10 m/s.

p=0.160×10=1.6 kg m/sp = 0.160 \times 10 = 1.6\ \text{kg m/s}.

The answer carries kg m/s, the unit that the bracket [MLT⁻¹] predicts.

Common slips on linear momentum

  • Writing [MLT⁻²]. That is force. Momentum has velocity in it, not acceleration.
  • Writing [ML²T⁻¹] for linear momentum. The extra length belongs to the angular quantity only.
  • Forgetting to convert grams and km/h. Dimensions stay the same, but the number is wrong unless the units are SI.
  • Thinking momentum and impulse must differ. They share one bracket; impulse is a change in momentum.

For other mechanics quantities, the dimensional formulas hub has each one with its own derivation.

Frequently asked questions

Is momentum a scalar or a vector?

Momentum is a vector, and its direction is that of the velocity. Its dimensional formula, [MLT⁻¹], says nothing about direction.

What dimensions does linear momentum have?

Written out with every power, [M¹ L¹ T⁻¹]: mass [M] multiplied by velocity [LT⁻¹].

Which SI unit is used for momentum, and is N s allowed?

The standard unit is kg m/s. It can also be written N s, because a newton second reduces to the same base units.

Do momentum and impulse have the same dimensions?

Yes. Impulse is force times time, [MLT⁻²] × [T], which gives [MLT⁻¹], the same as momentum.

How do the dimensions of momentum and force differ?

Force has one extra power of time in the denominator. Momentum is [MLT⁻¹], force is [MLT⁻²], because force is the rate of change of momentum.