kinematics, t0 is zero

With $a(t) = a$, $v(t_0) = v_0$, $x(t_0) = x_0$, $t_0 \ne 0$, the equations are
$$\tag{H1} v(t) = v_0 + a(t - t_0), $$ $$\tag{H2} x(t) = x_0 + v_0 (t - t_0) + \tfrac12 a (t - t_0)^2. $$
Set $t_0 = 0$, that makes $v(0) = v_0$, $x(0) = x_0$, will turn them into
$$\tag{H3} v(t) = v_0 + at, $$ $$\tag{H4} x(t) = x_0 + v_0 t + \tfrac12 a t^2. $$