## Need help with a speed/time/distance problem [closed]

I'm not any sort of math wiz, and I've run up against a problem that is fairly complex for me to solve. A friend suggested this site might be able to provide some help. So let me try to describe the issue as best I can. Let me start out by saying that I had prepared a couple of images to help explain all this, but I'm not allowed to use them in this post as I'm a new user. Hence, some references to graphs are less meaningful. I have tried to describe what the graphs depicted.

I have a path of a known distance, that must be traversed in a fixed amount of time. However, I must start the traversal of the path and end the traversal at a specific speed. So, for example, if I need to traverse 1200 feet in 10 seconds, and my entry & exit speeds must both be 120 ft./sec, then I can simply stay at the constant speed of 120 ft./second to accomplish my goal. If I graph speed against time, the area under the graph represents distance traveled as so:

(Image deleted due to being a new user. It showed speed in the vertical axis, time in the horizontal axis, with points marked for 120 ft./sec. on the vertical and 10 seconds on the horizontal. It showed a rectangular area under the horizontal line at Speed 120 ft/sec. starting a 0 seconds and going until 10 seconds. The area shown represents the 1200 feet that would be traversed).

However, if I have to travel only 700 feet in that same 10 second interval, things get ugly. I thought about decelerating at a constant rate until I could then accelerate at a constant rate to end up with my speed curve carving a triangle out of the graph in Figure 1 above, whose area above the curve would be 500 ft. However that would yield a discontinuity in the acceleration/deceleration that is unacceptable.

I then figured I could use a segment of a circle to do the same thing as shown below:

(Cool image deleted that shows a similar graph to the one deleted above, but with a segment of a circle cutting into the shaded area from the deleted image above, such that the segment intersected horizontal line at time = 0 and speed = 120 ft/sec on one side and 120 ft./sec and 10 seconds on the other side, with the segment dipping down to carve out 500 "feet" from the area under the horizontal line representing a constant speed of 120 ft/sec)

Here the orange area would represent the 500 ft less than the distance traveled by a constant speed. Following the speed curve indicated by the circle segment should be pretty trivial. And so it would seem that I have solved my problem. However, when I try to actually implement this into an algorithm, I run into the problem that the area calculations for the segment of a circle doesn't seem to yield units that make any sense. Perhaps it would be better to say that I don't know how to set up the problem so that the units make sense. Sure I can calculate the area of the segment, but what does 10 seconds mean when used as the chord of the circle, and what should the units of the radius be. I guess the value of theta is still easy at least. ;) Unfortunately I'm sort of stumped on the rest. I'm not even sure that this approach is viable.

I'd be just as interested in a numerical approach to the solution as a mathematical approach.

Any help you can offer to help me get my head around this would be greatly appreciated.

= Ed =

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This qustions seems like a very good one for math.stackexchange.com a similar site but with a broader scope. Here it is a bit out off scope, as the site is for (advaced) graduate mathematics and question related to current research in mathematics. – quid Dec 8 2011 at 0:43
Seconding quid's comment. I also took the liberty of removing the algebraic geometry tag, which is really meant for something different than the problem you describe – Yemon Choi Dec 8 2011 at 0:46
(Also, I think that math.stackexchange.com doesn't have minimum requirements for posting images, so hopefully you'll be able to use them.) – BR Dec 8 2011 at 0:54