# Solve Max Velocity given Distance, Time, Initial velocity, Acceleration, Deceleration

For a motor application I wan't to be able to find the maximum velocity I should ask my motor to use knowing:

• Distance to go D
• Time to travel T
• Acceleration a
• Deceleration d
• Initial velocity v0

So that my motor will go from initial position p0 with initial velocity p0 to final position with velocity = 0.

I found several post doing it the other way. Solving for time given maximum velocity. And I have a hard time reversing the equation.

Thank you for your help.

Ben

• Can you write down the solution for time given maximum velocity? If possible, do it by editing your post and using MathJax. The link for reference is here: math.meta.stackexchange.com/questions/5020/… – 5xum Nov 13 '18 at 9:23

If the vehicle accelerates from velocity $$v_0$$ to velocity $$v_{max}$$ over a time $$T_1$$, then we have

$$T_1 = \frac{v_{max} -v_0}{a}$$

If it then decelerates from velocity $$v_{max}$$ to rest over a time $$T_2$$, then we have

$$T_2 = \frac{v_{max}}{d}$$

We know that the total time $$T$$ must equal $$T_1+T_2$$, so

$$T = \frac{v_{max} -v_0}{a} + \frac{v_{max}}{d}$$

Re-arranging this to solve for $$v_{max}$$ gives:

$$v_{max} = \frac{d}{a+d}(aT+v_0)$$

• Thank you for your help! – I33N Nov 14 '18 at 20:24