Dimensional Formula of Acceleration [LT⁻²] with Derivation

In bracket notation the dimensional formula of acceleration reads [LT⁻²], or [M⁰ L¹ T⁻²] when every base quantity is shown. Acceleration involves length once and time twice in the denominator, and mass does not appear.

The two powers of time are the whole story. Velocity already divides a length by a time; acceleration divides that velocity by another time. This page builds the bracket from a = (v − u)/t, reads off m/s², lists the quantities that share it, checks a kinematics equation and covers the usual errors. Solved motion problems sit with the acceleration formula, next to its rearranged forms.

Quick facts

Dimensional formula
[M⁰ L¹ T⁻²], short form [LT⁻²]
Defining formula
$a = \frac{v - u}{t}$
SI unit
m/s²
Vector or scalar
Vector
Same dimensions as
g, centripetal acceleration, N/kg
Chapter
Motion in a straight line, Class 9 and 11

How the dimensional formula of acceleration is built

Use the defining relation, where u (written v₀ or vᵢ in many US books) is the initial velocity and v the final one.

$a = \frac{v - u}{t}$ .
A difference of two velocities is still a velocity: $[v - u] = [L T^{-1}]$ .
Time: $[t] = [T]$ .
Divide: $[a] = [L T^{-1}] \div [T]$ , which gives $[L T^{-2}]$ , or $[M^{0} L^{1} T^{-2}]$ in full.

Subtraction never changes dimensions, because you can only subtract like from like. That is why the bracket of v − u is just the bracket of v, worked out on the velocity page.

From the bracket to metres per second squared

Replace L with metre and T⁻² with per second squared: m s⁻², usually written m/s². Mass carries the power zero, so no kilogram appears in the unit.

The unit reads as "metres per second, every second": the speed gained or lost in each second. Retardation, or deceleration, is the same quantity with a negative sign, so it has the same unit and the same dimensions.

A car goes from rest to 72 km/h in 10 s. Find its acceleration in SI units.

Convert the speed first: 72 × 1000/3600 = 20 m/s.

$a = \frac{20 - 0}{10} = 2.0\ \text{m/s}^2$ .

The unit m/s² is exactly the bracket [LT⁻²] written with SI base units.

Accelerations in disguise

Quantities that share [M⁰ L¹ T⁻²]
Quantity	Formula	Comment
Free fall acceleration g	$g = \frac{GM}{R^{2}}$	A particular acceleration
Acceleration in a circle	$\frac{v^{2}}{r}$	[L²T⁻²] ÷ [L]
Gravitational field strength	$\frac{F}{m}$	N/kg is the same as m/s²
Tangential acceleration	$a_t = r\alpha$	Length times [T⁻²]

Gravitational field strength is the surprising row. Force per unit mass, N/kg, reduces to m/s², so field strength and free fall acceleration are the same quantity viewed two ways. See the acceleration due to gravity page.

Angular acceleration α is not on this list. It is [T⁻²], with no length, measured in rad/s².

Testing a kinematics equation

Check: v² = u² + 2as

Left side: $[v^{2}] = [L^{2} T^{-2}]$ .

Right side: $[u^{2}] = [L^{2} T^{-2}]$ , and $[a][s] = [L T^{-2}][L] = [L^{2} T^{-2}]$ .

All three terms carry the same bracket, so the equation is dimensionally consistent. The 2 is a pure number and cannot be tested this way.

Try the same test on s = ut + ½at². Each term should reduce to [L]: ut is [LT⁻¹][T] and ½at² is [LT⁻²][T²]. Both give length, as they must.

Typical mistakes

Writing [LT⁻¹]. That is velocity. Acceleration divides by time once more.
Putting mass in. [MLT⁻²] is force. Acceleration has M⁰.
Giving retardation a different bracket. A sign is not a dimension.
Confusing a with α. Linear acceleration has a length; angular acceleration does not.

The displacement to force ladder

Acceleration is the middle rung of a short ladder. Displacement is [L]. Divide by time for velocity, [LT⁻¹]. Divide again for acceleration, [LT⁻²]. Multiply by mass for force, [MLT⁻²].

Graph reading follows the same ladder. On a v against t graph the slope has units of (m/s)/s, an acceleration, and the area under an acceleration time graph is a change in velocity. The dimensional formulas page carries the ladder on to energy and power.

Frequently asked questions

Is acceleration a vector?

Yes. It points along Δv, the change in velocity. The dimensional formula is the same whatever the direction.

What is the dimensional formula for acceleration in full form?

In full it is [M⁰ L¹ T⁻²]. Most books shorten it to [LT⁻²] because powers of zero and one are often left out.

Why does time appear squared in acceleration?

Acceleration is velocity per time, and velocity is already length per time. Dividing by time twice gives T⁻².

Does retardation have the same dimensions as acceleration?

Yes. Retardation is negative acceleration, so it is also [LT⁻²] with the unit m/s².

Which SI unit goes with the acceleration bracket?

It is m/s²: metres for the length, and seconds squared below the line for the T⁻².