A random thought came into my head today when I was in the subway:
Suppose we have a train in a subway where the stations are evenly spaced in a straight line. The train accelerates for some amount of time, moves with a constant speed for some amount of time, and starts decelerating until it reaches the next station such that the same amount of time spent accelerating and decelerating are the same, and the magnitudes of both are the same. Create a possible function that illustrates this scenario.
I decided that I would try to use some combination of a sinusoidal function and a linear function, since the train would try to do the same thing during certain periods of time, and the train is moving only in one direction. I hopped onto Desmos and played around a little bit, and I was able to create a function that gets me somewhat close to what I want. $$d_1(t)=t-\frac1{2\pi}\sin 2\pi t$$
I chose the constant multiple $\frac1{2\pi}$ to simplify the distance between each station to $1$, and the multiplier $2\pi$ for $t$ to suggest that it takes $1$ unit of time to get from one station to the next. Choosing the sinusoidal function means that there will be no length of time where the train is moving at a constant speed, and the acceleration and deceleration will be exactly opposite, separated by the point of inflection halfway between the stations.
The problem: trains usually stop at every station. I want to, arbitrarily at first, modify the function so that the train waits for the same amount of time as it takes to travel from one station to another. Eventually, I want to be able to construct a function that can describe the waiting time in some proportion to the traveling time. I came up with the function
$$d_2(t)=\begin{cases}t-\frac{\lfloor t\rfloor}2-\frac1{2\pi}\sin 2\pi t,& \lfloor t\rfloor \textrm{ is even}\\ \frac{\lceil t\rceil}2,& \lfloor t\rfloor \textrm{ is odd}
\end{cases}$$
This gets me something closer to what I want.
My questions:
- What alternative ways of representing the scenario are there? For example, would it be possible to construct a piecewise function based only on polynomials that meets the criteria above?
- Is there a way to represent this particular scenario without use of a piecewise function?
- Suppose there must be a stretch of time where the train is moving at a constant speed between stations. How can I factor that in to the equation for the function?
- Suppose the train makes the return trip in a similar manner, and it makes some number of return trips daily. Does this suggest that it can be written as a periodic function? If so, what would be the equation for that function?
Just collecting various comments all in one place, and filling in some holes / details.
First, a function $d(t)$ is simply a mapping from values in the domain (in your case: time) to values in the range (in your case: distance). Whether you can write that function as a "nice" formula of "well-known" expressions, is kinda irrelevant. I should add though, the confusion between (abstract) functions and (nice) formulas is historic, so if you have the same confusion, you're not alone. :)
In the context of $d(t)$ being the motion of a train, there are reasonable assumptions to be adopted, e.g. $d$ should be continuous (the train does not "teleport"), differentiable (velocity is well defined) and maybe twice differentiable (acceleration is well defined). However, even after adopting these, there is nothing special about sinusoids. (In the context of a moving pendulum, or (idealized) planetary rotation, maybe sinusoids would be natural, but a moving train...?)
Q1: As answered by @79037662 it is surely possible to model the train motion $d(t)$ as piecewise polynomial - and in fact for a train this would probably be more natural than modeling by sinusoids. Assuming no friction (ha!) a zero acceleration would give constant velocity $v(t)$ and linear $d(t)$, a constant non-zero acceleration would give linear velocity $v(t)$ and quadratic $d(t)$, a linear acceleration would give quadratic $v(t)$ and cubic $d(t)$, etc. If you have pieces of acceleration you will have pieces of polynomial $d(t)$ and once you have pieces of polynomials you simply have to be careful to "glue" them together. The piecewise formula might be tedious to write out but there is nothing magical going on.
Q2: As answered by @quarague it is not possible for your function to have a constant stretch and also be analytic, and "analytic" very roughly translates to something with a nice formula (and probably includes most "familiar" functions you have in mind: polynomials, sinusoids, exponentials, etc). This is actually a non-trivial result. This applies to both distance (constant distance implying stopping at a station) and velocity: If you have constant stretches and non-constant stretches (in either distance or velocity), then you will have to write your function as piecewise.
Q3: Constant speed simply means that part of $d(t)$ is linear, e.g. $d(t) = v t + C$. Here $v$ is the constant velocity. You get this $d(t)$ by integrating $v$ over $t$.
Q4: As answered by @AdamRubinson Fourier Series would be a way to go, but for the purpose of writing out the function it's probably overkill. Suppose $d^*(t)$ for $t \in [0,1]$ describes what happens to the train in the first hour, and $d^*(0)=d^*(1)$ s.t. the train makes a complete round trip. Further suppose the train does this every hour, then you can simply write the overall $d(t)$ as
$$d(t) = d^*(t - \lfloor t \rfloor)$$
Here $\lfloor t \rfloor$ is the standard notation for the "floor" function which denotes the greatest integer $\le t$, so e.g. if $t=13.7$ then $\lfloor t \rfloor = 13$ and $t - \lfloor t \rfloor = 0.7$ gives the fraction part of $t$. Note that while "floor" is a "standard" function, it is piecewise. Which in a roundabout way brings us back to the your Q2: you can always describe a piecewise function in a non-piecewise way... if you define a new name for it! :)
Hope this helps (as opposed to confuses even further!)