Acceleration and speed from timed events

Question

Basically I have data from a list which shows how an automated scenery truck should move. The guys in the who have given it have basically given me an acceleration time and deceleration (or negative acceleration) time and a time at full speed they also have told me how far the piece has traveled. I have converted the imperial data into metric which gives the travel distance in this example as $8509mm$.

Our system uses acceleration and speed in $mm/s^2$ and $mm/s$ respectively and therefore I need to find a way of converting the US times into accelerations and speeds. In a separate document they have mentioned that in this example the full speed would be $1,215.644 mm/s$ so based on that I can assume the acceleration is $v/t$ so $607.822 mm/s^2$ and the neg accel is $405.214 mm/s^2$.

The trouble is we do not have this full speed number quoted in the 2nd document for every move and therefore I need to derive this full speed from what I was given (accel time, full speed time, decel (neg accel) time and total distance).

The example data I have is:

Acceleration = $2s$

Deceleration = $3s$

Full Speed for = $4.5s$

Total move time = $9.5s$

Distance = $8509 mm$

What would be the formula for calculating the acceleration and "deceleration" in $mm/s^2$ and the top speed?

Answer 1

You are given acceleration time $t_1$, deceleration time $t_2$, full speed time $t_f$, and the total distance $x$. You have three unknowns, the acceleration $a_1$, the deceleration $-a_2$, and the full speed $v_f$. Therefore you need three independent equations.

The distance traveled while accelerating is $$x_1=\frac12a_1t_1^2$$ The distance traveled at full speed is $$x_f=v_ft_f$$ The distance traveled while decelerating is $$x_2=v_ft_2-\frac 12 a_2t_2^2$$ The total distance traveled is $$x=\frac12a_1t_1^2+v_ft_f+v_ft_2-\frac 12 a_2t_2^2$$

You can relate acceleration and deceleration to the full speed: $$v_f=a_1t_1=a_2t_2$$ You should be able to solve the system now.

For example, use $a_1=v_f/t_1$ and $a_2=vf/t_2$. Then $$x=v_f\left(\frac12t_1+t_f+\frac12t_2\right)$$ Your only unknown here is $v_f$.

Answer 2

Thanks so much Andrei

So basically i wound up taking:

$x$ = $v_f$((0.5*$t_1$) + $t_f$ + (0.5*$t_2$)) and rearranged to:

$v_f$ = $x$ / ((0.5*$t_1$) + $t_f$ + (0.5*$t_2$))

Using that I was then able to use the following to find:

$a_1$ = $v_f$ / $t_1$

and

$a_2$ = $v_f$ / $t_2$

(again please forgive some of the "workarounds" for the mathematical notation).

Thanks for persevering with me!