The International System of Units — Question 3

Question 3

A unit check can reject a formula, but can it prove one?

A cart starts from rest. For this idealized experiment its acceleration $a$ is constant and its displacement after elapsed time $t$ is known independently to obey $$x=\tfrac12 at^2.$$ Use $[x]=\mathrm{m}$, $[a]=\mathrm{m\,s^{-2}}$, and $[t]=\mathrm{s}$. Three software routines propose $$\text{A: }x=at,\qquad \text{B: }x=at^2,\qquad \text{C: }x=at^2+b.$$

1. Which routines can be rejected by units alone? What unit must $b$ have for C to pass a unit check?

2. With $a=2.0\ \mathrm{m\,s^{-2}}$ and $t=3.0\ \mathrm{s}$, compare B with the stated physical relation. Explain what this establishes about unit checks.

3. Suppose C uses one fixed value of $b$. Can it reproduce the stated relation both at $t=0$ and at $t=3.0\ \mathrm{s}$? Justify without assuming that dimensional consistency guarantees the answer.

This is an exercise in evaluating equations, not a request to derive the motion law. Treat the given values as exact within the model.

Question 3 - Solution

A unit check can reject a formula, but can it prove one?

(a) Check products and sums separately.

$$[at]=\mathrm{m\,s^{-1}},\qquad[at^2]=\mathrm m,\qquad[b]=\mathrm m.$$

Routine A gives $[at]=\mathrm{m\,s^{-1}}$, a speed unit rather than a displacement unit, so A fails. In B, $[at^2]=\mathrm{m}$, so B passes a unit check.

For C, the first term has unit metre. Quantities being added must have compatible dimensions; hence $\boxed{[b]=\mathrm{m}}$. With this choice C passes the unit check, although its physical validity remains untested. The number zero can represent a zero displacement; it does not require adding a new kind of quantity.

(b) Test the numerical coefficient.

The supplied physical relation gives $$x=\tfrac12(2.0)(3.0)^2\ \mathrm{m}=\boxed{9.0\ \mathrm{m}},$$ whereas B gives $18.0\ \mathrm{m}$. The factor $1/2$ has no units, so unit analysis cannot detect its absence. Dimensional consistency is necessary for a physically meaningful equation in this form, but it is not sufficient to establish the physical law.

(c) Test one parameter at two times.

$$b\big|_{t=0}=0,\qquad b\big|_{t=3\,\mathrm s}=9\ \mathrm m-18\ \mathrm m=-9\ \mathrm m.$$

At $t=0$, the stated displacement is zero. Routine C then requires $b=0\ \mathrm{m}$. At $t=3.0\ \mathrm{s}$, C would consequently predict $18.0\ \mathrm{m}$, not $9.0\ \mathrm{m}$.

Alternatively, fitting only the latter time would require $b=-9.0\ \mathrm{m}$, which fails at $t=0$. No single constant $b$ satisfies both conditions. A constant offset cannot correct this time-dependent error in the coefficient of $t^2$.