Dimensional analysis: checking ½mv²=mgh with exponent vectors and ruling out v=2gh

Goal and scope

With integer exponent vectors d(Q)=(a,b,c) over the base dimensions L, M, T: compute the dimensions of speed, acceleration and two energy expressions; decide that ½mv²=mgh is dimensionally consistent, that v=2gh is not and cannot be repaired by a pure-number constant, and that v²=4gh is consistent but wrong in the constant-acceleration model; decide that m+v cannot be added; solve E=C·m^α v^β for α=1 and β=2; and explain that equal dimensions do not mean equal unit conversion factors.

Uses only the three base dimensions L, M, T and integer exponents (MKO assumption a1); ½, 2 and 4 are declared dimensionless, and g is a dimensioned quantity (assumption a2). The constant-acceleration model used in S11 is stated in G3 and is not derived by dimensional analysis. SI conversion details, Buckingham Pi and nondimensionalization are not treated; the canonical MKO and its English edition are not changed.

Givens and symbols

• cartesian_product_def
• numeric_evaluation
• eq_congruence
• power_laws
• distributive

• G1 — Base dimensions L, M, T; the dimension [Q]=L^a M^b T^c of a quantity Q is represented by the integer vector d(Q)=(a,b,c) (MKO assumption a1).
• G2 — d(m)=(0,1,0), d(h)=(1,0,0), d(t)=(0,0,1); speed v is length divided by time and acceleration g is speed divided by time; ½, 2, 4 and C are dimensionless pure numbers (assumption a2).
• G3 — Constant-acceleration model: starting from rest with acceleration g>0, after time t>0 the speed is v=gt and the distance travelled is h=gt²/2.
• G4 — Expressions to check: (E1) ½mv²=mgh; (E2) v=2gh; (E3) v²=4gh; (E4) the sum m+v.

• d(Q)=(a,b,c) — exponent vector of the quantity Q, [Q]=L^a M^b T^c
• m,\ h,\ t — mass, height, time
• v,\ g — speed, acceleration
• E — energy (kinetic)
• C,\ \alpha,\ \beta — dimensionless constant and unknown exponents

Rules and sources

• cartesian_product_def · prerequisite — Cartesian product: A×B is the set of ordered pairs (a,b) with a∈A and b∈B; (a,b)=(c,d) ⇔ a=c and b=d.
• numeric_evaluation · prerequisite — Exact evaluation of an explicit integer or rational expression; independently recomputable.
• eq_congruence · prerequisite — Congruence of equality: if u=v, applying the same defined operation f to both sides gives f(u)=f(v). Operating on one side only is not an instance of this rule.
• power_laws · prerequisite — Integer exponent laws (base nonzero, or exponents positive integers): a^m·a^n=a^{m+n}, (a^m)^n=a^{mn}, (ab)^n=a^n b^n, a^0=1.
• distributive · prerequisite — Distributivity: u(v+w)=uv+uw; with commutativity also (v+w)u=vu+wu.
• def_intro · prerequisite — Introducing a definition: a statement, expression or object may be given a name, after which the name and what it stands for are interchangeable.
• add_zero · prerequisite — Additive identity: u+0=u.
• contradiction · prerequisite — Proof by contradiction: if assuming ¬P leads to a contradiction (a statement and its negation both hold), then P holds.
• mul_assoc · prerequisite — Associativity of multiplication: (uv)w=u(vw).
• mul_comm · prerequisite — Commutativity of multiplication: uv=vu (in commutative number systems such as ℤ, ℚ, ℝ, ℂ).
• square_nonneg · prerequisite — Squares are nonnegative: for every real u, u²≥0, and u²=0 ⇔ u=0.
• order_basics · prerequisite — Order of the reals: exactly one of u<v, u=v, u>v holds; u<v and v<w give u<w; u<v gives u+w<v+w; u<v and w>0 give uw<vw; u<v and w<0 give uw>vw.
• and_intro_elim · prerequisite — Conjunction: from P and Q infer P∧Q; from P∧Q infer P and infer Q.
• mko_dim · cited — Relative to basic dimensions L,M,T, write [Q]=L^a M^b T^c, represented by (a,b,c). Products add vectors, quotients subtract them and integer powers scale them. Quantities to be equated or added must have matching dimensions. · mko-dimensional-analysis
• mko_dim_limits · cited — Dimensional consistency is necessary but not sufficient for a physical equation; if the two sides have different dimensions, adjusting a pure-number constant cannot fix it, and if they agree there may still be a wrong coefficient. Dimensional analysis does not track the numerical factors of unit conversion. · mko-dimensional-analysis
• vec_ops · prerequisite — Componentwise operations in ℤ³: (a,b,c)±(a′,b′,c′)=(a±a′,b±b′,c±c′) and k(a,b,c)=(ka,kb,kc); two vectors are equal exactly when the three components are equal; a pure number has dimension vector (0,0,0).
• L_const_factor · lemma — For a dimensionless constant k and any quantity Q, d(kQ)=d(Q).

Main derivation

[Q]=L^aM^bT^c

d(PQ)=d(P)+d(Q),\ d(P/Q)=d(P)-d(Q),\ d(Q^k)=k\,d(Q)

m,\ h,\ t

d(m)=(0,1,0),\ d(h)=(1,0,0),\ d(t)=(0,0,1)

d(v)=d(h)-d(t)

d(v)=(1,0,-1)

d(g)=d(v)-d(t)

d(g)=(1,0,-2)

d(v^2)=2\,d(v)

d(v^2)=(2,0,-2)

d(\tfrac{1}{2}mv^2)=d(\tfrac{1}{2})+d(m)+d(v^2)

d(\tfrac{1}{2}mv^2)=(2,1,-2)

d(mgh)=d(m)+d(g)+d(h)

d(mgh)=(2,1,-2)

d(\tfrac{1}{2}mv^2)=(2,1,-2),\ d(mgh)=(2,1,-2)

\text{(E1) dimensionally consistent}

d(v)=(1,0,-1),\ d(2gh)=(2,0,-2)

\text{(E2) inconsistent};\ d(kgh)=(2,0,-2)\ \forall k\ \text{dimensionless}

d(v^2)=(2,0,-2),\ d(4gh)

d(4gh)=(2,0,-2):\ \text{(E3) dimensionally consistent}

v=gt,\ h=\tfrac{1}{2}gt^2

v^2=2gh,\ \text{so }v^2=4gh\ \text{fails for }g,t>0

m+v,\ d(m)=(0,1,0),\ d(v)=(1,0,-1)

d(m)\neq d(v):\ m+v\ \text{not allowed}

(2,1,-2)=\alpha(0,1,0)+\beta(1,0,-1)

\alpha=1,\ \beta=2,\ E=C\,m\,v^2

\mathrm{J}=\mathrm{kg\,m^2\,s^{-2}},\ \mathrm{erg}=\mathrm{g\,cm^2\,s^{-2}}

d=(2,1,-2)\ \text{for both},\ 1\,\mathrm{J}=10^7\,\mathrm{erg}

\text{proved facts}

\text{(E1) consistent; (E2) inconsistent; (E3) consistent but false in G3; (E4) not allowed; }E=Cmv^2

Substeps and lemmas

d(kQ)

d(kQ)=d(k)+d(Q)=(0,0,0)+d(Q)=d(Q)

v^2=(gt)^2,\ 2gh=2g\cdot\tfrac{1}{2}gt^2

v^2=g^2t^2=2gh

v^2=4gh\ \text{(assumed)}

v^2=2v^2\Rightarrow v^2=0\Rightarrow v=0,\ \text{contradicting }v=gt>0

Reverse audits

• S15 · sufficient_goal_reduction · S08, S09, S10, S11, S12, S13, and_intro_elim — Goal G: the determinations in the goal. Sufficient condition H: S08 to S13, each already proved (H⇒G by conjunction introduction). H is derived forwards from the givens; G is never used as a premise.
• S14 · premise_audit · mko_dim_limits, power_laws, numeric_evaluation — It needs the definitions of the two unit systems and powers of 10; the conclusion is illustrative and independent of the earlier dimensional verdicts.
• S13 · inverse_equivalence · vec_ops, cartesian_product_def — Vector equality ⇔ the three components are equal; substituting α=1, β=2 gives (2,1,−2), so both directions hold.
• S12 · premise_audit · S02, S03, mko_dim, vec_ops — It needs only that the two vectors differ.
• S11.2 · premise_audit · S11.1, square_nonneg, order_basics, contradiction — It needs g, t>0; at t=0 (before any motion) v=h=0 and both relations hold trivially, which gives no support to (E3).
• S11.1 · inverse_equivalence · power_laws, mul_assoc, mul_comm — Exponent laws and commutativity and associativity of multiplication; both sides are the same expression.
• S11 · premise_audit · G3, mko_dim_limits — It needs the stated physical model and g, t>0; dimensional analysis alone cannot supply this verdict.
• S10 · premise_audit · L_const_factor, vec_ops, mko_dim — As in S08, it only passes the necessary condition; S11 shows that it is still wrong.
• S09.1 · inverse_equivalence · vec_ops, add_zero — An identity of adding the zero vector, valid both ways; it holds only for dimensionless k, since a dimensioned constant would change the vector.
• S09 · premise_audit · L_const_factor, vec_ops, mko_dim_limits — Unequal vectors need only one differing component. The conclusion is that (E2) is certainly wrong; what the correct relation is is not claimed here and needs a physical model (S11).
• S08 · premise_audit · S06, S07, mko_dim_limits — The conclusion is only 'passes the necessary condition'; the converse (consistent ⇒ correct) is not claimed.
• S07 · inverse_equivalence · vec_ops, numeric_evaluation — Componentwise integer addition, valid both ways.
• S06 · inverse_equivalence · vec_ops, numeric_evaluation — Componentwise integer addition; both sides are the same vector.
• S05 · inverse_equivalence · vec_ops — Multiplying by 2 and halving undo each other on these even components, so both ways hold.
• S04 · inverse_equivalence · vec_ops, numeric_evaluation — Componentwise integer arithmetic, valid both ways.
• S03 · inverse_equivalence · vec_ops, numeric_evaluation — Componentwise integer arithmetic; backwards, (1,0,−1)+(0,0,1)=(1,0,0) restores it.
• S02 · premise_audit · G2, def_intro — Taken directly from the givens; no new fact is derived.
• S01 · premise_audit · G1, mko_dim, power_laws — This step only states the rules and derives no new fact; exponents are restricted to integers.

Error controls and negative controls

Dimensional consistency does not make an equation correct

One misconception named by the MKO: both sides of v²=4gh are (2,0,−2), but in the constant-acceleration model the correct relation is v²=2gh (S11). Dimensional analysis cannot detect a wrong dimensionless coefficient, a wrong direction or a wrong physical assumption; it can only rule out inconsistent expressions.

A unit name is not a dimension exponent

Another misconception named by the MKO: the joule and the erg have the same dimension vector but differ by a factor of 10⁷ (S14). Notation also needs care: the m in the formulas is mass (dimension M), while the unit 'm' is the metre (dimension L); they must not be confused.

Pure numbers cannot fix an inconsistency; dimensioned constants must be declared

By L_const_factor, multiplying by any dimensionless k leaves the dimension unchanged, so v=k·gh is inconsistent for every k (S09). Conversely, g is a dimensioned quantity (1,0,−2); treating it as a pure number would spoil the check of (E1). MKO assumption a2 requires the dimensions of constants to be stated.

• v²=4gh is dimensionally consistent, so it is the correct free-fall relation → reject — The constant-acceleration model gives v²=2gh; if v²=4gh also held then v=0, contradicting v=gt>0 (S11.2).
• With the right dimensionless constant k, v=k·gh becomes dimensionally consistent → reject — d(k·gh)=(2,0,−2)≠(1,0,−1)=d(v), whatever k is (S09, L_const_factor).
• m+v is a meaningful quantity with dimension (0,1,0)+(1,0,−1)=(1,1,−1) → reject — Only quantities with the same dimension can be added; adding vectors corresponds to the product mv, not the sum (S12).
• The joule and the erg have the same dimension, so 1 J=1 erg → reject — 1 J=10⁷ erg; dimensional analysis does not track conversion factors (S14).

Result and limitations

d(v)=(1,0,−1), d(g)=(1,0,−2), and d(½mv²)=d(mgh)=(2,1,−2), so ½mv²=mgh is dimensionally consistent. v=2gh has sides (1,0,−1) and (2,0,−2), which differ, and no dimensionless constant can repair it. v²=4gh is consistent, but in the constant-acceleration model v²=2gh, so it is false: consistency is only necessary. m and v have different dimensions and cannot be added. From (2,1,−2)=α(0,1,0)+β(1,0,−1), α=1 and β=2, so E=C m v² with C not fixed by dimensions. The joule and the erg have the same dimension but 1 J=10⁷ erg.

• The constant-acceleration model of S11 (v=gt, h=gt²/2) is used as a given; this guide does not derive it from mechanics.
• SI conversion, the Buckingham Pi theorem and nondimensionalization belong to the MKO's formal next step.

Verification and source binding

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