Piecewise
Position $$\tag{D3} x(t) = \left\{ \begin{array}{cc} x_0 + v_0(t - t_0) + \tfrac12 a_0(t - t_0)^2, & t_0 \le t < t_1, \\[0.5em] x_1 + v_1(t - t_1) + \tfrac12 a_1(t - t_1)^2, & t_1 \le t < t_2, \end{array} \right. $$ with $t_{n+1} = t_n + \tau_n$, where $\tau_n$ is $n$-th time interval. Initial conditions $x(t_0) = x_0$, $x(t_1) = x_1$ and $x(t_1) = x_0 + v_0(t_1 - t_0) + \tfrac12 a_0(t_1 - t_0)^2$.Velocity $$\tag{D2} v(t) = \left\{ \begin{array}{cc} v_0 + a_0(t - t_0), & t_0 \le t < t_1, \\[0.5em] v_1 + a_1(t - t_1), & t_1 \le t < t_2, \end{array} \right. $$ with $t_{n+1} = t_n + \tau_n$, where $\tau_n$ is $n$-th time interval. Initial conditions $v(t_0) = v_0$, $v(t_1) = v_1$ and $v(t_1) = v_0 + a_0(t_1 - t_0)$.Acceleration $$\tag{D1} a(t) = \left\{ \begin{array}{cc} a_0, & t_0 \le t < t_1, \\[0.5em] a_1, & t_1 \le t < t_2, \end{array} \right. $$ with $t_{n+1} = t_n + \tau_n$, where $\tau_n$ is $n$-th time interval.Acceleration $$\tag{C1} a(t) = a_n, \ \ \ \ t_n < t < t_{n+1}. $$ Velocity $$\tag{C2} v(t) = v_n + a_n(t - t_n), \ \ \ \ t_n \le t \le t_{n+1}. $$ Posisition $$\tag{C3} x(t) = x_n + v_n(t - t_n) + \tfrac12 a_n(t - t_n)^2, \ \ \ \ t_n \le t \le t_{n+1}. $$Velocity $$\tag{B1} v(t) = v_n + a(t - t_n), \ \ \ \ t_n \le t \le t_{n+1}. $$ Initial condition $$\tag{B2} x(t_n) = x_n. $$ Posisition $$\tag{B3} x(t) = x_n + v_n(t - t_n) + \tfrac12 a_n(t - t_n)^2, \ \ \ \ t_n \le t \le t_{n+1}. $$Constant acceleration $$\tag{A1} a(t) = a_n, \ \ \ \ t_n < t < t_{n+1}. $$ Initial condition $$\tag{A2} v(t_n) = v_n. $$ Velocity $$\tag{A3} v(t) = v_n + a_n(t - t_n), \ \ \ \ t_n \le t \le t_{n+1}. $$The inspiration comes while in the trip using subway. Assume that visited 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. What would be a possible function that illustrates this scenario?A piecewise-constant acceleration is constant over extended time intervals and changing in value discontinuously from one interval to the next. This is physically unrealistic in any real-life situation, since the acceleration would be expected to change more or less smoothly from instant to instant. In realistic models of collisions the acceleration changes smoothly, but it still can be approximated as continuous for simplification.