Inverse proportionality intuition

Often in physics we model something using proportionality constants. When we can intuitively say that some quantity grows as another quantity grows, we might assume that the relationship is linear and that the quantities correlate somehow as follows: $$a = kb$$ Where $a$ and $b$ are quantities and $k$ is a constant number.

Now we can measure some pairs of $a$ and $b$, and divide $a$ by $b$ to get the constant $k$. We can then proceed to compute other unknown values of $a$ by multiplying $k$ with other $b$ values if we can indeed assume that the correlation is linear.

But when quantity $a$ seems to decrease as $b$ increases, we often assume a different model:

$$a = k/b$$

Why do we often tend to assume that $a$ is proportional to the inverse of $b$ instead of there being a negative $k$?

• This sounds like a very strange question. We don’t just guess random equations based on whether things are increasing or decreasing as others change. There are actual reasons the equations take the form they do. – knzhou Aug 26 '18 at 20:49
• If somebody literally told you “since x increases when y decreases, they must be inversely proportional”, then they were completely wrong. It’s a total non sequitur. – knzhou Aug 26 '18 at 20:49
• You can guess what you want, but the lab rules. Practically everything is carefully checked experimentally. – FGSUZ Aug 26 '18 at 21:02
• You don't assume anything. You look carefully at the data and back-calculate the value of k that matches that data. – David White Aug 27 '18 at 4:54
• I think it is most for the appearance of the formula, because if something is up and increases then if something decreases we put it down. Like as if our minds thinks "ok this time is the contrary, it decreases" so it looks for something to invert. But instead of the sign of k it thinks of the position. It's probably more imminent. – Costantino Aug 27 '18 at 20:49

Linear models and "inverse linear" models would be relevant if they are either exact fits or "good enough" approximations. In that case they will often simplify the equations enough to allow an analytical solution. Without those simplifications an analytic solution is often not practical or possible in a useful closed form.

A very well known example where we use a linear approximation to get a relatively easy solution form would be the analysis of the pendulum. There we use the small angle approximation to get a simple linear equation that's easy to work with as an approximation.

Note there are many ways to have $y$ decrease as $x$ increases. The only requirement is :

$$\frac {dy}{dx} < 0$$

And there are literally no end of functions that can fit that. Perhaps the most common to see is an exponential function like this $y=e^{-x}$ and related functions.

Some help can sometimes be found in using dimensional analysis, as per this example or this post, which may help you obtain relations between exponents of quantities through the use of dimensionless variables. In many practical situations, one would make a plot of $a$ vs $b$ and fit the resulting curve or discrete set of experimental points with some “guess” curve. For instance, the curve may suggest that $$a=k b^c$$ so that the exponent $c$ and the constant $k$ can be found from a log-log graph of the data.

Of course in the end if insufficient data is available and there is no theory to suggest a functional relation between $a$ and $b$, then one is completely in the dark.