I am currently working on a project where I model the motion of a snowboarder using a 4th-degree polynomial. The polynomial is currently defined in abstract units along the x-axis and y-axis. I have encountered a challenge when attempting to convert these abstract units into real-world time (seconds on x-axis and meters in y-axis) to make the model's predictions more interpretable and align with real-world measurements.
The specific context is as follows:
The polynomial $P(x)$ models the motion as a function of x abstract units.
I have determined that a horizontal distance span of 16.9 abstract units in the model corresponds to 2.70 seconds in real-world time. This was measured from video footage of the snowboarder, marking two points along their path.
Given this, I initially thought to convert the x-axis from abstract units to time by directly applying the ratio as a scale factor, aiming to express each unit of $x$ in terms of real-world time $t$, writting the equivalency as $18x=2.7t$. The direct approach led me to believe that $x = \frac{2.7t}{16.9}$ would correctly scale the model. However, graphical interpretations and further considerations suggest that the inverse, $x = \frac{16.9t}{2.7}$, might be more appropriate for accurately representing the model within the expected real-world timeframe.
Here are my questions and doubts:
How should I correctly derive the scale factor for converting abstract units into real-world time in this context? Is the direct approach ($x = \frac{2.7t}{16.9}$) or the inverse ($x = \frac{16.9t}{2.7}$) more appropriate for modeling purposes, and why?
Are there any mathematical principles or modeling considerations that favor one conversion method over the other in scenarios where the relationship between abstract measurements and real-world quantities is established through empirical observation? Am I overlooking any factors?
I appreciate any insights or guidance!