Raptor problem #2

[realMakc]

here. for the sake of discussion, we have (let's take seemingly missing data from problem #1):

raptors attack accelerating at 4 m/s^2 up to their top speed of 25 m/s. you're at the center of equilateral triangle with a raptor at each corner. the top raptor has a wounded leg and is limited to a top speed of 10 m/s. you begin to flee, quickly (1) reaching your top speed of 6 m/s. the raptor will run towards you. at what angle (2) you should run to maximize the time you stay alive?

three problems with that is (1) what does "quickly" mean (∞ m/s^2 ?), then (2) who said that straight line is the optimal route to take, and (3) should everyone keep running at their top speed once it was reached?

how would you go about solving this problem? I am tempted to write simple "genetic" solver based on brute-force simulation.

[newblack]

i think it's a really complicated problem because how the raptors react and accelerate toward your changes in movement is hard to define. but if you assume that they don't lose speed on changing direction there might be an analytical way of solving it. i think it might be a type of quadratic programming problem? where you have to maximize a convex set given certain constraints.

if you picture this geometrically, where rn is a raptor, r1 is the injured one and x is you:

    r1

     x

r2      r3 

the solution is the point q where the graphs of r1, x and r3 (or r2, it doesn't matter) intersect that yields the greatest area of the set { r1, x, r3, q }.

if x, r1 and r3 abstract the position, velocity and accelerations of each element you're trying to maximize the function f(x, r1, r3) subject to the constraints that ||x vel|| <= 6 m/s, ||r1 vel|| <= 10 m/s and ||r3 vel|| <= 25 m/s.

but i'm just guessing! i don't really know, but i'm curious how you end up solving it!

also, my intuition tells me that the deviation you should run from the path perpendicular to the line segment (r1,r3) is equivalent to the ratio of the raptors' relative maximum speeds....

[realMakc]

the solution is the point q where the graphs of r1, x and r3 (or r2, it doesn't matter) intersect that yields the greatest area of the set { r1, x, r3, q }

imagine injured raptor totally disabled (v1 = 0) then solution would be q on (x-r2)+(x-r3) line, or r1-x line actually, as long as q != r1, which does not maximize that area; that also suggests that solution must depend on velocities... even if we throw out initial acceleration to simplify whole thing a bit, it is not quite easy to solve.