Dimensional Formula of Velocity and Speed [LT⁻¹] Explained

Velocity carries one power of length and one inverse power of time, so the dimensional formula of velocity is [M⁰ L¹ T⁻¹], or [LT⁻¹] in short. Speed has exactly the same bracket.

Velocity is displacement per unit time, v = s/t. Speed is distance per unit time. One is a vector and the other a scalar, but both divide a length by a time, and dimensions ignore direction. This page derives the bracket, explains the unit m/s, converts km/h, lists every speed that shares [LT⁻¹] and tests two equations. Numerical problems are handled with the velocity formula itself.

Quick facts

Dimensional formula
[M⁰ L¹ T⁻¹], short form [LT⁻¹]
Defining formula
$v = \frac{s}{t}$
SI unit
m/s
Speed
Same dimensions as velocity
Vector or scalar
Velocity is a vector, speed a scalar
Conversion
1 km/h = 5/18 m/s

Building the dimensional formula of velocity from v = s/t

Whether a question asks for the dimension formula of velocity or the dimensional formula for velocity, the working is the same, and only two base quantities appear.

Definition: $v = \frac{s}{t}$ , displacement over time taken.
Displacement is a length: $[s] = [L]$ .
Time is a base quantity: $[t] = [T]$ .
Ratio: $[v] = [L][T^{-1}] = [L T^{-1}]$ ; with mass shown, $[M^{0} L^{1} T^{-1}]$ .

The dimension formula of speed follows the same three lines with distance in place of displacement. Average velocity, instantaneous velocity ds/dt and relative velocity all share it, because a derivative of a length with respect to time is still a length over a time.

The unit m/s and the km/h conversion

Swap L for metre and T⁻¹ for per second. The SI unit is m s⁻¹, written m/s. Because mass has power zero, no kilogram appears.

Convert 90 km/h to m/s using the dimensions

Velocity has L¹ and T⁻¹, so multiply by the length factor once and divide by the time factor once.

$90\ \text{km/h} = 90 \times \frac{1000\ \text{m}}{3600\ \text{s}} = 25\ \text{m/s}$ .

The shortcut ×5/18 is just 1000/3600 reduced. Going the other way, multiply m/s by 18/5.

If a converted speed looks a thousand times too big, check that you divided by 3600 and did not multiply.

Every kind of speed has this bracket

Every one of these is a speed in m/s
Quantity	Formula	Comment
Speed of light	$c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$	A fixed constant, 3.00 × 10⁸ m/s
Wave speed	$v = f\lambda$	[T⁻¹] × [L]
Escape velocity	$v_e = \sqrt{2gR}$	Square root of [L²T⁻²]
Orbital speed	$v_o = \sqrt{\frac{GM}{r}}$	Also a square root of [L²T⁻²]
Drift velocity of electrons	$v_d = \frac{I}{neA}$	Very slow, but still m/s

Whenever a square root shows up in a speed formula, the quantity under it must be a speed squared, [L²T⁻²]. That habit catches a missing factor before you calculate. The speed of light page shows the μ₀ε₀ case in detail.

Two equations put through a dimension test

Check: v = u + at

$[u] = [L T^{-1}]$ and $[a t] = [L T^{-2}][T] = [L T^{-1}]$ .

Both terms on the right match the left side, so the first equation of motion is consistent.

Find the error in v = √(2g/h)

Inside the root: $\frac{[L T^{-2}]}{[L]} = [T^{-2}]$ . The root gives $[T^{-1}]$ , not $[L T^{-1}]$ .

So the formula is wrong. The correct one, v = √(2gh), has [L²T⁻²] under the root, which gives a speed.

Where answers go wrong

Writing [LT⁻²]. That is acceleration. Velocity has one division by time, not two.
Thinking speed and velocity have different dimensions. Direction is not a dimension.
Using [LT] for velocity. Time is in the denominator; its power is −1.
Converting km/h by multiplying by 3.6. That goes from m/s to km/h, the opposite way.

Quantities built on velocity

Velocity is the first derived quantity most students meet, and many others are built on it. Multiply by mass for momentum, [MLT⁻¹]. Divide by time for acceleration, [LT⁻²]. Square it and multiply by mass for kinetic energy, [ML²T⁻²].

Follow those steps on the momentum and acceleration pages. The complete table of derived brackets sits on the dimensional formulas page.

Frequently asked questions

What is the dimensional formula of speed, and does it match velocity?

It matches exactly: each is length divided by time, [M⁰ L¹ T⁻¹]. Velocity also has a direction, but dimensions do not record direction.

Is velocity a vector?

Yes. Velocity is a vector; speed, its size, is a scalar. Both are measured in m/s.

Which unit does the bracket [LT⁻¹] give in SI?

Metre per second, m/s. It is the length unit divided by the time unit, with nothing else.

Why is the power of mass zero in velocity?

Neither displacement nor time involves mass. A feather and a truck at the same speed have the same velocity in m/s.

How do the dimensions of velocity and acceleration differ?

Acceleration has one more division by time. Velocity is [LT⁻¹], acceleration is [LT⁻²].