How fast can it be?

Suppose Brilli starts with velocity $v_0$ , decelerates at rate $a_1$ until time $t_1$ and then accelerates at rate $a_2$ from $t_1$ to $t_f$ . The final distance is $x_f$ , giving the following relation.

$\large{\frac{1}{2} (v_0 + v_1) t_1 + \frac{1}{2} (v_1 + v_f) (t_f - t_1) = x_f \\ v_1 = v_0 - a_1 \, t_1 \\ v_f = v_1 + a_2 \, (t_f - t_1)}$

Our task is to find $t_1, a_1, a_2$ such that the following are satisfied:

1) The equations above hold true
2) The quantity $v_f$ is maximized
3) Accelerations are no larger than $1$

Performing this optimization yields:

$\large{a_1 = a_2 = 1 \\ t_1 \approx 0.2929 \\ v_f = \sqrt{2} \approx 1.4142}$