Accelerate to Max velocity, then decelerate to known velocity

Question

I have an object traveling at a known velocity (Vi). It then accelerates (known A) to a known maximum velocity (Vmax), then decelerates (-A) to another known velocity (Vf). The total distance traveled is also known (Xf-Xi). I'm looking for an equation that will give me total elapsed time (t) when Xi, Xf, Vi, Vmax, Vf, and A are all known quantities. It also needs to take into account that Vmax may not be attained if the (Xf-Xi) is too small.

Answer 1

The three important formulas in kinematics are $$ d=v_it + \frac12at^2$$ $$a=\frac{v_f-v_i}t$$ $$v_f^2 = v_i^2 + 2ad$$ where

$d =$ travelled distance

$v_i =$ initial velocity

$v_f =$ final velocity

$a =$ acceleration

$t =$ elapsed time

Can you proceed?