If ship A and ship B start out traveling in a straight line with ship B trailing, any change in direction by ship A with an immediate response from ship B will result in a loss of distance between A and B. Every time the direction of A is changed, B is given an opportunity to cut the corner.
I guess that would depend on your definition of “instant.”
Then again…it was defined at the outset.
No. it does not matter. If both ships have the exact same speed then their distance is maintained when the first travels directly away from the second and diminishes in any other case. It never increases. Since it decreases at times it follows that the pursuer will catch up.
And if it is instant, it neither increases nor decreases.
No, it does not matter if it is instant or not. A change in course of A results in B closing in. If it does not result in B closing in then it was not a change in course.
So, you are saying that ship B gets a better “instant” than ship A does? If that’s the case, the game is rigged.
Ship B doesn’t have to move in the same direction as ship A.
Edit: There’s a book that deals with this sort of problem. I haven’t read it, but the reviewers seem to think it’s decent.
WTF?
B travels in a straight line towards A. If A travels away then distance is maintained. The instant A changes direction distance diminishes. If A is forced to change direction then distance diminishes.
Exactly. The instant A changes course, things change. If the instant B changes course things also change, then B never catches A. That’s what I said in the beginning.
You are wrong.
That’s rather a compelling argument.
Doh! For some reason I was picturing 90 degree turns only, like on graph paper, who knows why.
I find it amusing that the names of so many posters to this thread are related to the topic.
Let’s write out the details of sailor, et al,'s correct point:
Suppose ship B is constantly moving directly towards A, at a speed of S. And A is constantly moving at a speed of S, though there may be some changes of direction involved (even instantaneous discontinuous changes of direction).
So these are the two factors playing into the rate of change of the distance between A and B. The first factor (the velocity of B) clearly contributes to the rate of change of this distance a term -S. As for the second factor (the velocity of A), we can, at any particular moment, split this into a linear combination of the vector from B to A and a perpendicular vector; the latter will contribute nothing to the (momentary) rate of change of the distance, while the former will contribute to this rate by an amount exactly equal to its magnitude, being either an increase or a decrease depending on whether it is in the direction from B to A or the reverse. But its magnitude can be at most the magnitude of A’s velocity, which is S. Thus, it contributes at most an increase of S to the distance between B and A [this only happening in the case where A is moving in the direction directly away from B].
Therefore, at any given moment, the total rate of change in the distance between A and B is -S + [something which, at most, is S]. At most, this is 0, while this drops to a negative whenever, as noted above, A is not moving in the direction directly away from B. Which is exactly sailor’s point: whenever A is not moving in the direction directly away from B, B is gaining ground on A. Thus, if A is forced to make changes of direction, even instantaneous ones, then A is forced to, at some points, move in a direction not directly away from B (that is, if A was always moving in the direction directly away from B, then it would never be changing direction). And during those times when A is not moving directly away from B, B must be gaining ground on A.
That having been said, it doesn’t follow that B will necessarily eventually catch up; just that it will keep getting closer.
Expanding on that last line: That having been said, it doesn’t follow that B will necessarily eventually catch up, or even that it will do so asymptotically; this just guarantees that it will keep getting closer. It seems easy enough to construct situations where B keeps getting closer but never gets closer than X% of its starting distance, for any appropriate X.
I beg to differ. Presuming that there’s a minimum value of a turn that qualifies as a turn, so that we’re not dealing with something like 1/infinity degrees, each turn is going to create a real decrease in distance. That real value might shrink each time, but as it does it approaches zero, making it an infinitesimal and so giving you a hard value: Time of Impact. If I can calculate the ToI, then most certainly the one does catch the other in all cases.
As I showed in my post, in all cases, A will be traversing the outside of a circle. B will be traversing an expanding spiral on the inside of that circle. Since their speed is the same, B has a shorter distance to travel and will intersect at a specific time.
Very basic proof using triangles.
Imagine the segment between the ships as the base. The chaser wants to take a heading that would complete the third side of an isosceles triangle, therefore his heading must be the same angle to the base as the ship being chased.
HOWEVER, if the ship being chased is traveling 90 degrees or more to the base, the chaser will never catch it since no isosceles triangle can be formed.
The distance can keep decreasing without reaching 0 or anything close to it. E.g., suppose the distance goes from 1 to 0.95 to 0.905 to 0.9005 to 0.90005 to 0.900005 to 0.9000005… This keeps decreasing, but never hits 0, or even gets below 0.9.
Your triangles are misguided; the base is the segment between the two ships, but what are the other two sides of the triangle supposed to correspond to? It looks like you want them to correspond to the velocities of these ships, since you demand that they have the same size. But why should these two culminate in the same point to serve as the third vertex? There’s no reason they should. Indeed, there’s no reason you should want to construct a triangle whose sides mix and match distances and velocities (things of different types, with different units) in the first place.
I think you are missing that B changes directions so that it can travel in a shorter line to meet A.