# Integrating equations with units

I was looking through an old copy of Barron's AP Physics and found this problem relating to impulse which I was initially confused about how to integrate.

Example 6.1 During a collision with a wall lasting from $t=0$ to $t=2\text{ s}$, the force acting on a $2\text{-kg}$ object is given by the equation $\mathbf{F} = (4\mathrm{\ kg\ m/s^4})t(2s-t)\hat{i}$

They work out that the integral is equal to:

$$\frac{16}{3}\hat{i}\frac{\text{kg m}}{\text{s}}$$

I am confused about the role of the units in the problem.

Looking at the answer, it seems that if I were to just ignore all the units and simply integrate $4t\cdot(2-t)$ that would give me $16/3$, and because that is a force I know it to be $\mathrm{kg\,m/s}$ or $\mathrm{N}$.

Why it would be OK to ignore the units in the integral though is somewhat unintuitive to me (other than that I know the end result must be a force) and I feel like it may get me into trouble with other problems.

Can someone explain how it is that the scalar values of the original unit quantities still give you the correct answer?

You know that you can pull a multiplicative constant out in front of an integral, right?

$$\int cf(t)\mathrm{d}t = c\int f(t)\mathrm{d}t$$

where $f(t)$ is any function of $t$, like $t^2$ or $t(2\text{ s} - t)$ (and $c$ does not depend on $t$).

Units can be part of that constant factor too. In this case, the constant factor is $4\mathrm{\ kg\ m/s^4}$.

The reason this all works is that an integral is basically an addition. You're computing the value of a function, in your case $4\mathrm{\ kg\ m/s^4}t(2\text{ s} - t)\hat{i}$ at some time $t$, multiplying it by a small increment in time $\mathrm{d}t$, and adding up the result for all possible times. Take a look at the units of the different pieces you add up:

\begin{align} &4\color{blue}{\mathrm{\ kg\ m/s^4}}\color{red}{t}\color{green}{(2\text{ s} - t)}\hat{i}\color{purple}{\mathrm{d}t} \\ \text{units: }&(1)\color{blue}{(\mathrm{kg\ m/s^4})}\color{red}{(\text{s})}\color{green}{(\text{s})}(1)\color{purple}{(\text{s})} = \frac{\text{kg m}}{\text{s}^4}\text{s}^3 = \frac{\text{kg m}}{\text{s}} \end{align}

So you're adding up things which have units of $\text{kg m/s}$. Thus, your result will have the same units. Since the units are a constant factor, it doesn't matter whether you pull them out in advance or leave them in for the integration.

This probably is why it is useful to use variables. If we just let $q=4\,{\rm kg\,m/s^4}$ and $h=2\,{\rm s}$, then your force is $$\mathbf{F}=qt\left(h-t\right)\hat{\mathbf{i}}$$ We then integrate this over time $t$, $$\int_0^t\mathbf{F}\,dt'=q\int_0^tt'\left(h-t'\right)dt'\,\hat{\mathbf{i}}$$ we get, $$\int_0^t\mathbf{F}\,dt'=q\frac{-2t^3-6ht^2}{6}\,\hat{\mathbf{i}}\Rightarrow\frac{16}{3}[q][t][h]\hat{\mathbf{i}}$$ where $[q]$ represents the units of $q$ and likewise for the other variables. Looking at these, we get $$\rm{\frac{kg\,m}{s^4}\cdot s^3=\frac{kg\,m}{s}}$$ which is the unit of impulse, not force.

It is my opinion, then, that the appropriate way is to wrap the units into a variable, integrate, then apply the units.

There does seem to be the tacit assertion \begin{align} & \int^{2s}_{0s} 4t(2s - t) \frac{kg·m}{s^4} 𝐢 dt \\ = & \int^{2s}_{0s} 4 \frac{t}{s}·s(2s - \frac{t}{s}·s) \frac{kg·m}{s^4} 𝐢 d\left(\frac{t}{s}·s\right) \\ = & \int^2_0 4 τ(2 - τ) \frac{kg·m}{s^4} s^2 𝐢 s dτ \\ = & \int^2_0 4 τ(2 - τ) \frac{kg·m}{s} 𝐢 dτ \\ = & \left(\int^2_0 4 τ(2 - τ) dτ\right) \frac{kg·m}{s} 𝐢 \\ = & \frac{16}{3} \frac{kg·m}{s} 𝐢 \end{align} that you can just freely stick in or pull out units from calculus operations as if they were constants. This Sapir-Whorfs the assertion "units are constant" directly into the language, itself. Contained in that assertion is the even deeper tacit assertion that a physical quantity, such as a "meter" at one point in space and time is comparable to a "meter" at another point in space and time; never mind whether that comparison is that they are the "same" or not.

The $$𝐢$$ part is already accounted for. In a general setting ... of a curvilinear coordinate chart in a space-time geometry that may or may not be flat, it is a vector in a "frame", and frames are represented as "vierbeins"; the gauging of which is part of the specification of a gravity-inertia field. A notion of "covariant derivative" arises for such geometric objects.

For units, there is no formalism that I am aware of that gauges units to provide a means to compare units at different places, and no corresponding notion of a "covariantly constant" unit ... other than the trivial one that's tacitly used that "a unit is constant".

Along with the idea of unit-gauging is the notion that the "sameness" scaling may be path-dependent, if the underlying gauge space is non-trivial. So, you could have two observers who start up together and end up together, who each get copies of the same units at their starting point, who each carry units that "remain the same" along their respective paths, but who come back together with units that no longer match - an Arbitrage Of Units, you might say.

You have a similar issue with commodities and currencies if they're not all pegged to a common standard, like the dollar, which is why you have a common standard. The possibility of Unit Arbitrage is Sapir-Whorfed out of the language by the "units are constant" assertion built into the language.

Another possibility that's quietly hidden away is the notion of an Infinite King's Thumb Problem, where the unit is a quantity that goes to 0 or infinity somewhere. That would show up as a "singularity" in the quantities measured with respect to those units. For instance, of the meter were tied to light-seconds in a system of units and light speed went to infinity (say, at the beginning of time), then it would show up as spatial distances all going to zero.

There are attempts at creating formalisms for unit scaling. For instance, in

"Gauge Field Theory in Natural Geometric Language: A Revisitation of Mathematical Notions of Quantum Physics 9780198861492, 0198861494"

https://dokumen.pub/gauge-field-theory-in-natural-geometric-language-a-revisitation-of-mathematical-notions-of-quantum-physics-9780198861492-0198861494.html

you'll see an attempt at one in section 2.1 ("Unit Spaces And Physical Scales"); and I've seen a similar formalism used in a paper at the 2023 RaMiCS earlier this year. But if you look carefully at it, you'll see that the "unit is constant" assertion was quietly sneaked into the definition of the differential operator, by slipping the unit space out from under the differential operator. There was no attempt at implementing any gauging of units in the formalism.