# Accelerate to Max velocity, then decelerate to known velocity

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.

• Does it travel any distance at $V_i$ before accelerating? Does it travel any distance after decelerating to $V_f$? If the answer to both questions is No, then you can easily calculate the time as $\frac{V_{max}-V_i}{A}+\frac{V_{max}-V_f}{A}$. If the answer to both is Yes, then the total time is indeterminate. – almagest Jun 10 '16 at 18:00

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?