Which statement about the equation s = ut + at² is correct?

It is dimensionally correct but physically wrong

It is dimensionally wrong and physically wrong

It is dimensionally correct but physically wrong

It is both dimensionally and physically correct

Its correctness cannot be judged dimensionally

Units, Dimensions and Error Analysis

This is the least glamorous chapter in Physics and reliably one of the most profitable: the exam asks one or two questions from it almost every year, they are usually solvable in under a minute, and the dimensional method it teaches will save you on some later question where you have forgotten a formula. There is a second reason to take it seriously. Every numerical answer you write in Section B has a unit and a significant-figure requirement, and marks are lost there not from bad physics but from a length left in centimetres. So treat this chapter as the discipline that protects every other chapter.

The SI Base Units and Dimensional Formulae

Seven base quantities generate everything else. Their dimensional symbols are [M] mass, [L] length, [T] time, [A] current, [K] temperature, [mol] amount of substance, [cd] luminous intensity. Every derived quantity is a product of powers of these, and that product is its dimensional formula.

Dimensional formulae worth knowing cold
Quantity	Dimensional formula	SI unit
Force, weight, tension	[M L T⁻²]	newton (N)
Work, energy, heat, torque	[M L² T⁻²]	joule (N·m)
Power	[M L² T⁻³]	watt (J/s)
Pressure, stress, modulus	[M L⁻¹ T⁻²]	pascal (N/m²)
Momentum, impulse	[M L T⁻¹]	kg·m/s = N·s
Planck constant, angular momentum	[M L² T⁻¹]	J·s
Gravitational constant G	[M⁻¹ L³ T⁻²]	N·m²/kg²
Coefficient of viscosity η	[M L⁻¹ T⁻¹]	poiseuille (Pa·s)
Surface tension, force constant	[M T⁻²]	N/m
Frequency, angular velocity, decay constant	[T⁻¹]	s⁻¹

🧠 Quantities that share a dimensional formula

A favourite question type asks which of four quantities has the same dimensions as a fifth, so learn the families rather than the entries. Work, energy, heat, torque and moment of a force are all [M L² T⁻²]. Pressure, stress, energy density and all elastic moduli are all [M L⁻¹ T⁻²]. Angular momentum, Planck's constant and the action share [M L² T⁻¹]. Impulse and momentum share [M L T⁻¹]. Frequency, angular frequency, velocity gradient and decay constant are all [T⁻¹]. Note carefully that torque and work have identical dimensions but are physically different — a scalar and a vector — which is precisely why dimensional analysis alone can never prove a formula is right.

ℹ️ Dimensionless quantities, and the ones with no unit but a name

Angle (radian), solid angle (steradian), strain, refractive index, relative density, Poisson's ratio, coefficient of friction, dielectric constant, Reynolds number and all pure ratios are dimensionless. Two consequences are examined. First, anything that appears as the argument of sin, cos, log or an exponential must be dimensionless — that fact alone solves a whole class of questions where you are asked to find the dimensions of a constant inside an exponent. Second, a dimensionless quantity has the same numerical value in every system of units, which is why refractive index and strain need no conversion.

What Dimensional Analysis Can and Cannot Do

The principle of homogeneity is the whole engine: in any valid physical equation every additive term has the same dimensions. From that single idea come three uses and two hard limits.

The three uses, and the two limits that questions exploit
Use / limit	How it works
Check an equation	If the dimensions of two added terms differ, the equation is certainly wrong
Convert between systems	n₁[M₁ᵃL₁ᵇT₁ᶜ] = n₂[M₂ᵃL₂ᵇT₂ᶜ] — the physical quantity is the same, so the numbers scale inversely to the unit sizes
Derive a relation	Assume Q ∝ aˣbʸcᶻ, equate powers of M, L, T on both sides, solve for x, y, z
Limit 1 — no constants	Dimensionless factors are invisible to the method, so it gives T = k√(l/g), never the 2π
Limit 2 — no sums, and no distinction of same-dimension quantities	It cannot produce s = ut + ½at², and it cannot tell torque from work

🧠 The derivation that is asked, worked once

Find how the period T of a simple pendulum depends on length l, mass m and g. Assume T ∝ lˣ mʸ gᶻ and write dimensions: [T] = [L]ˣ[M]ʸ[LT⁻²]ᶻ. Match each base separately. M: 0 = y, so the period cannot depend on mass at all — which is a real physical result, obtained without any experiment. T: 1 = −2z, so z = −½. L: 0 = x + z, so x = +½. Hence T ∝ √(l/g), and experiment supplies the 2π that dimensions cannot. Two habits make this fast in the exam: always equate the base with the fewest occurrences first (M here, since it appears once), and remember that a zero exponent is itself an answer worth stating.

⚠️ Dimensionally correct does not mean correct

Questions test this directly. s = ut + at² is dimensionally flawless and physically wrong, because the ½ is missing — and dimensions cannot see a pure number. Likewise v² = u² + 2as and v² = u² + 3as are both dimensionally sound. So the logic runs one way only: dimensional failure proves an equation wrong, dimensional success proves nothing. Read the option wording carefully, because a question asking "which is dimensionally correct" and one asking "which is correct" can have different answers among the same four options.

Errors — How They Combine

Two ideas cover the whole of this topic. Absolute error Δa is the uncertainty in a measurement; relative (fractional) error is Δa/a and the percentage error is 100 Δa/a. Which one propagates and how depends only on whether the quantities are added or multiplied.

The four propagation rules — this table is the chapter
Operation	Rule	Note
Z = A + B or A − B	Absolute errors add: ΔZ = ΔA + ΔB	They add even for subtraction — errors never cancel
Z = A × B or A / B	Relative errors add: ΔZ/Z = ΔA/A + ΔB/B	Again addition, whether multiplying or dividing
Z = Aⁿ	ΔZ/Z = \|n\| · ΔA/A	A square doubles the error; a square root halves it
Z = AᵖBᑫ/Cʳ	ΔZ/Z = p(ΔA/A) + q(ΔB/B) + r(ΔC/C)	All terms positive — the worst case is what is reported

🎯 Why the largest power dominates, and what to measure carefully

Because errors enter multiplied by their exponents, the quantity raised to the highest power dominates the uncertainty. In the Young's modulus relation Y = 4MgL/(πd²x), a 1% error in the diameter d contributes 2%, while a 1% error in L contributes 1%. So the practical instruction is to measure the highest-power quantity most carefully — which is exactly why the diameter of a wire is measured with a screw gauge (least count 0.01 mm) while its length is fine with a metre scale. Questions are frequently set as "which measurement should be made most precisely", and the answer is always the one with the largest exponent, not the one with the smallest value.

⚠️ Significant figures — the two rules that differ

For multiplication and division, the result keeps as many significant figures as the least precise factor: 4.237 × 2.1 = 8.9, not 8.8977. For addition and subtraction, the result keeps as many decimal places as the term with fewest: 12.11 + 8.2 = 20.3. These two rules are genuinely different — one counts significant figures, the other decimal places — and confusing them is a standard error. Two further points: trailing zeros after a decimal point are significant (2.30 has three), and a change of unit never changes the count, so 2.30 m and 230 cm and 0.00230 km all carry three significant figures. This matters directly in Section B, where a question may specify the number of decimal places for your typed answer.

Key takeaways

Seven base dimensions: M, L, T, A, K, mol, cd. Every derived quantity is a product of their powers.
Work, energy, heat and torque all share [M L² T⁻²]; pressure, stress and all moduli share [M L⁻¹ T⁻²].
Anything inside sin, cos, log or an exponential must be dimensionless — that alone solves a class of questions.
Dimensional failure proves an equation wrong; dimensional success proves nothing (s = ut + at² is dimensionally fine).
Dimensional analysis cannot supply dimensionless constants — it gives T ∝ √(l/g), never the 2π.
Add or subtract → ABSOLUTE errors add. Multiply or divide → RELATIVE errors add. Both, even for the inverse operation.
For Z = Aⁿ, ΔZ/Z = |n|ΔA/A, so measure the highest-power quantity most precisely.
Multiplication keeps the least significant figures; addition keeps the least decimal places — different rules.

Practice questions (8)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

Which of the following does NOT have the dimensions [M L² T⁻²]?
1. Torque
2. Work
3. Power
4. Heat
Show answer
Answer: C — Power
Power is energy per unit time, so it is [M L² T⁻³] — one extra power of T⁻¹. Torque, work and heat all share [M L² T⁻²]. Note that torque and work having identical dimensions while being physically different (a vector and a scalar) is exactly why dimensional analysis can never prove a formula correct.
In the equation y = A sin(kx − ωt), the dimensions of k are:
1. [L]
2. [L⁻¹]
3. [T⁻¹]
4. Dimensionless
Show answer
Answer: B — [L⁻¹]
The argument of a sine must be dimensionless, so kx is dimensionless and k must be [L⁻¹] to cancel the [L] of x. By the same reasoning ωt is dimensionless, giving ω = [T⁻¹]. This one principle — that anything inside sin, cos, log or an exponential is dimensionless — solves an entire family of questions.
Which statement about the equation s = ut + at² is correct?
1. It is dimensionally wrong and physically wrong
2. It is dimensionally correct but physically wrong
3. It is both dimensionally and physically correct
4. Its correctness cannot be judged dimensionally
Show answer
Answer: B — It is dimensionally correct but physically wrong
Both ut and at² have dimensions [L], so the equation is dimensionally homogeneous — yet it is wrong, because the factor ½ is missing and dimensions cannot detect a pure number. This is the standard illustration that dimensional failure disproves an equation while dimensional success proves nothing.
If Z = A²B / C³, and A, B, C carry percentage errors of 1%, 2% and 1% respectively, the percentage error in Z is:
1. 4%
2. 7%
3. 1%
4. 3%
Show answer
Answer: B — 7%
Relative errors add, each multiplied by the magnitude of its exponent: 2(1%) + 1(2%) + 3(1%) = 2 + 2 + 3 = 7%. The exponent of C is 3 even though it is in the denominator — all terms enter positively, because the reported error is the worst case and errors never cancel.
Young's modulus is found from Y = 4MgL/(πd²x). Which quantity must be measured most precisely?
1. The mass M
2. The length L
3. The diameter d
4. The extension x
Show answer
Answer: C — The diameter d
The diameter appears squared, so its relative error is doubled in the result while M, L and x contribute their error once each. That is precisely why a wire's diameter is measured with a screw gauge of least count 0.01 mm while its length is adequately measured with a metre scale. The answer is always the largest exponent, not the smallest value.
Z = A − B, where A = 10.0 ± 0.2 and B = 8.0 ± 0.3. The result is:
1. 2.0 ± 0.1
2. 2.0 ± 0.5
3. 2.0 ± 0.25
4. 18.0 ± 0.5
Show answer
Answer: B — 2.0 ± 0.5
For a sum or a difference the absolute errors add: 0.2 + 0.3 = 0.5. They add even though the quantities are subtracted, because errors never cancel — the tempting 0.1 comes from subtracting them. Note also how bad this measurement is: an absolute error of 0.5 on a value of 2.0 is a 25% relative error, which is why subtracting two nearly equal quantities is avoided in experiment design.
The result of 4.237 × 2.1, expressed to the correct number of significant figures, is:
1. 8.8977
2. 8.90
3. 8.9
4. 9
Show answer
Answer: C — 8.9
In multiplication the result carries as many significant figures as the least precise factor, and 2.1 has two, so the answer is 8.9. Note that the rule for addition is different — there the result keeps the least number of decimal places, so 12.11 + 8.2 = 20.3. Confusing the two rules is a standard error.
Using dimensional analysis for the period of a simple pendulum T ∝ lˣ mʸ gᶻ gives:
1. x = ½, y = 0, z = −½
2. x = ½, y = ½, z = −½
3. x = 1, y = 0, z = −1
4. x = −½, y = 0, z = ½
Show answer
Answer: A — x = ½, y = 0, z = −½
Matching [T] = [L]ˣ[M]ʸ[LT⁻²]ᶻ base by base: M gives y = 0, so the period cannot depend on mass — a real physical result obtained without experiment; T gives 1 = −2z so z = −½; L gives 0 = x + z so x = ½. Hence T ∝ √(l/g). Dimensions cannot supply the 2π, which is the method's fundamental limit.