Acc
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 $$\tag{A3} v(t) = v_n + a_n(t - t_n), \ \ \ \ t_n \le t \le t_{n+1}. $$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.