What is displacement time series? For a car motion, I have time vector (starting from 0 to 40 with 0.1 increments) so that the time vector is something like $\begin{bmatrix}0&0.1&0.2&0.3&\ldots&40\end{bmatrix}$ and I also have the corresponding velocity vector of the car. I am asked to generate the displacement time series using trapezoidal Area of the time/velocity data.
I have no idea what the displacement time series in this problem mean. Could you please tell me what is meant by the the displacement time series?
I know that displacement can be calculated by integrating velocity but I am not sure what the displacement time series means.
 A: The notation being followed here is cranky and misleading. The intended meaning for the exercise can be imagined this way - you are given a table which gives the instantaneous velocity $v(t)$ of the car at some specific times, all separated by fixed time increments of 0.1 units. As you mention in the question, you can obtain overall displacement by integrating the velocity with time. 
Very strictly speaking, $s = \int v dt$ (i.e. for infinitesimal increments), but this could be loosely approximated by the summation $\sum_i v(t_i) \delta t_i$ which you can approximate using the trapezoidal rule for numerical integration. That is what the exercise is asking you to do. 
But now imagine it this way - suppose instead of reaching $t = 40$ units directly, you go in steps. You know $s = 0$ when $t = 0$, and use the above datasets to find out the value of $s$ at $t = 0.1$ units first. So, now you know $s(t = 0)$ as well as $s(t = 0.1)$. Now, take another time increment to arrive at $s(t = 0.2)$. Iterating this procedure, to cover each individual time increment, you can arrive at a data set similar to the velocity data set, which bears the value of net displacement at time $t_i$, i.e. $s(t_i)$. This is what is being referred to as the displacement time series, i.e. the value of displacement as a function of time.
Hope that helps :)
