I am trying to model the displacement of a straight-line travel that has non-constant speed (e.g. 100 m sprint, straight-line horse racing, etc.). The model only needs to be plausible; it does not have to be realistic. The function should be strictly increasing, but with slowing down and speeding up. I tried:
$f(x) = x + sin(\alpha x)sin(\beta x)$
where $\alpha < 1$ and $\beta < 1$
For example, this is the plot of $f(x) = x + sin(0.6x)sin(0.8x)$:
Some issues I have:
- Is this a correct model? i.e. Is $f(x)$ really a strictly increasing function?
- How can I remove the "regularity" of the speed increases and decreases? If you look at the graph of the function above, you can see that the "waves" (slowing down and speeding up) happen at constant intervals. Is there a way to vary the intervals of these "waves"?
- Are there other functions I should look into?