I have an object which starts at velocity = 0. Accelerates to a known max speed and continues thereon at constant velocity (max speed).

My problem: How long does it take to reach a known destination?

I've visually described it like this: visual description of problem Where we have velocity on vertical-axis, and time on horizontal-axis. Up to where the lines intersect (let's call this time $t_M$, the blue line is the current velocity, after that, the red line. Equation for blue line is $v = a \cdot t$ where $a$ is acceleration, $v$ is velocity (vertical-axis) and $t$ is time (horizontal-axis). Equation for the red line it $v = M$, where $M$ is the max speed (known) and $v$ velocity.

Since the distance to travel is know, it should be as easy as adding up the area under the blue line, up until it equals the requested distance. Is there any way to do this without handling the areas before and after $t_M$ individually?

My current solution is:

$a$ (acceleration, known), $d$ (requested distance, known), $M$ (max speed, known)

$t_M = M / a$

$distanceDuringAccel = t_M * M / 2$ (the area under the blue line)

if distanceDuringAccel > d

We will not reach our destination during acceleration, add the area under the red line aswell


we will reach destination during acceleration. t = sqrt(2 * d / a)

The above works, but feels very clumsy. Is there a better way to address this?


closed as off-topic by Brandon Enright, JamalS, John Rennie, Kyle Kanos, Jim Jun 4 '14 at 14:39

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  • 1
    $\begingroup$ @Joshua , the area of trapezoid wont be a simple function but a combination of different functions in different ranges. In the end you will end up following the same procedure described above, which is a perfectly acceptable way to solve such problems $\endgroup$ – udiboy1209 Jun 3 '14 at 19:05
  • $\begingroup$ @udiboy1209 you should write that as an answer. $\endgroup$ – Neuneck Jun 3 '14 at 19:16
  • $\begingroup$ It's not homework.. :) $\endgroup$ – Alexander Olsson Jun 5 '14 at 20:34

You could draw a graph of displacement vs. time, and draw a line parallel to x-axis for the specified position to see if it cuts the graph.

The graph method will help in more complicated cases where the velocity profile is not a straight line, or it is negative for some regions.

But again to plot the graph, you will have to split the velocity graph into discrete integrable domains, and integrate.


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