2 ships theoretical question

Factual Questions
81

This is just childish. What is the difference to you because to me they are the same thing. Are we just playing with words. The “course” of a ship is the direction of its trajectory. Or do you have any other definition?

For someone so bent on picking on words you have totally missed the meaning of

this means the vessel can go from course C to course C’ instantly, not that course C’ is maintined for no time at all because, you see, that would be a really silly question to ask as it means there was no change in course.

The question of whether a a constant multiplied by zero equals anything but zero is not what the OP is asking. You are misinterpreting the OP.

82

[quote=“Omphaloskeptic, post:80, topic:500351”]

As Indistinguishable has pointed out, your argument says that B can continue to get closer; it does not say that B can eventually catch up. There are at least three situations that I think should be distinguished here:[ol][li]B can catch A in finite time;[]B asymptotically approaches A, with the distance between A and B approaching zero but never mathematically equal to zero at finite time;[]The distance between A and B asymptotically approaches some nonzero value.[/ol]You can merge (1) and (2) as “close enough,” though I think the OP wanted to distinguish between these two cases. Some trajectories for A allow B to meet A in finite time; some don’t. I think (3) can only happen if A’s course changes are allowed to become arbitrarily small.[/li][/QUOTE]
I agree with your distinction.

83

Among other possibilities, one very sensible meaning for “course” is the one you note here: the actual orientation of the track along which something is moving. “Direction” - the word used in the OP - could mean that, and could also mean heading; given the specific statement that “direction changes are instant” - something that’s not possible for a course - it makes sense to prefer the latter meaning.

Yet the terms of the OP can be met with zero time on the new course - by mean of another instantaneous change back to the original direction.

Where in the OP do you find a requirement that a new course must be maintained for at least some non-zero time and distance? Thus far your support for this contention - upon which your argument depends - comes from outside the scope of the OP.

Yet it does meet the conditions for a change in heading, which is what the OP requires.

Without question, what’s proposed in the OP is impossible, and thus can well be described as silly. That doesn’t prevent it being discussed.

I think my interpretation is quite strictly accurate. Feel free to quote the words in the OP that I have incorrectly interpreted.

As noted above, you find what’s proposed there to be impossible (you are right) and therefore choose to infer the additional unstated condition that a direction change must persist for some non-zero time (which is rather specifically not required).

84

[quote=“Omphaloskeptic, post:80, topic:500351”]

As Indistinguishable has pointed out, your argument says that B can continue to get closer; it does not say that B can eventually catch up. There are at least three situations that I think should be distinguished here:[ol][li]B can catch A in finite time;[]B asymptotically approaches A, with the distance between A and B approaching zero but never mathematically equal to zero at finite time;[]The distance between A and B asymptotically approaches some nonzero value.[/ol]You can merge (1) and (2) as “close enough,” though I think the OP wanted to distinguish between these two cases. Some trajectories for A allow B to meet A in finite time; some don’t. I think (3) can only happen if A’s course changes are allowed to become arbitrarily small.[/li][/QUOTE]
I accept the distinction and I agree with it. I think we can all agree (3) is not happening for any meaningful meaning of “change course” (or “ship direction”, to please those who do not know what “course” means).

I can see where case (2) is possible for instance if A is travelling in a circle and B describes an outgoing spiral which never reaches the circle described by A.

My intuition is that in most or all case the answer is (1) but to give a definitive answer we need to establish how A is constrained to act because just saying it is obligated to alter course is not very meaningful. Maybe at least establish a minimum change (30, 45, 60, 90 degrees?) and minimum frequency.

85

Xema, I am not going to waste any more time arguing silly semantics. Omphaloskeptic and others understand the question being asked and that is all I am dealing with.

86

Why do you insist on being particularly bullheaded about this? :rolleyes:

Direction is where you are pointed. If you can “point” straight ahead, then “point” to the left, then “point” straight ahead again, all without using any time (thus, instantaneous), you have changed “direction.” Your “course” may not have changed if you want to define “course” as meaning at least two points in a given direction (your definition), but your “direction” (which we can sample from a single point, as you well know (else, you would never have a “direction” if you were travelling in a circle).

And as I’ve stated, this makes the question in the OP meaningless, so let’s assume that we can substitute your concept of “course” instead, and ignore the pedantic discussion of “direction” as opposed to “course” or “vector” or any other method of describing what the OP asked, unless the OP makes clear that he/she was attempting to fool you into the exact situation under discussion, in which case it stops being a problem in math and one of semantics.

87

Both.

You are just substituting a limit of time for a limit of distance. If R is the remaining difference in distance from B to A at each time interval, then you gave the limit R +1/2R, plus 1/4R, … I suppose you could take the pedantic view that the two ships will become infinely close in finite time but never touch. But in the real world the ships would collide as soon as the difference between them is small enough to be affected by any small change in current, wind, temperature of the ship materials, etc.

Or, a sailor decides to reach out and touch the other ship and the Captain says “don’t make me turn this ship around”.

88

So what? Dollars to donuts, the OP wasn’t asking this question as a practical matter, but as a mathematical puzzle. It wasn’t really meant to be about ships and water current, temperature, whatever. After all, they did label it “theoretical question”, not “need answers ASAP”.

89

I cannot see why this would be pedantic. The OP reads to me as a thought, theoretical, question, not as something intended to be practical. Most college questions are similarly formulated and arguing that they are not real life scenarios or over silly semantics will only get you to fail the course.

I think the OP is meant as a theoretical case where it is intended that we make abstraction of many real life factors. It seems many people are incapable of centering on the question asked and making abstraction of some real life constraints. An “infinite sea”? Imagine that!

This is not a contest to find out how things can be misinterpreted so that they are impossible or meaningless. Some people may enjoy doing that but it is just threadshitting.

90

How unfortunate that you stop just short of demonstrating my error by quoting something from the OP that shows I’m wrong.

In truth, we don’t disagree. You find the OP’s precise premise silly, and thus wish to discuss a variant of the problem where direction changes must persist for a finite time. I find the OP’s premise silly and show this by noting that under the specified rules you can change direction without actually altering course.

I’ll note that your variant of the problem retains some silliness (e.g. instant perception of direction changes by the following ship).

91

I’ll offer an example to try to convince you that (1) isn’t all that common. Let’s suppose that A changes heading exactly once every unit time, by an angle of either +t or -t, so that A’s course is a path formed from unit-length segments, each making an angle of ±(180°-t) with the last. To make it sporting, A always chooses to turn away from B: that is, if B is to the left of A’s heading line, A turns right; if B is to the right, A turns left. Let’s take t<90° so that A doesn’t make any very sharp turns.

I hope it’s intuitively reasonable that B’s optimum strategy, once B gets sufficiently close to A, is to aim straight for the point on A’s trajectory one unit away each time A changes heading, unless B is actually able to intercept A within this unit time. (If B aimed for a point one unit away and (say) to the left of A’s trajectory, A would turn right and B would not have gained as much ground.)

If you accept this strategy for B, then you can draw a triangle to find out how much B gains on A in each time step. If A and B were initially a distance x apart, with B directly behind A before A turns by angle t, then after one unit of time B can get to the point directly behind A and a distance
x’ = x cos t + 1 - sqrt(1 - x[sup]2[/sup] sin[sup]2[/sup] t) ;
for small x this can be approximated as
x’ ~= x cos t ,
which is intuitively sensible.

This means that once A and B get pretty close, in each following unit time interval B can only manage to cut the distance by a factor of cos t. This is a geometric sequence, which asymptotically approaches 0 but does not actually reach it at finite time (regardless of what Dan Blather has to say).

92

I find your interpretation of the OP does not follow from the wording and does not make sense from the intended meaning.

This thread, as often happens, has turned into a pointless semantic argument in which I am not interested.

93

You don’t seem to understand why the silliness is there. A problem like this often includes some non-real factors in order to relieve you from factoring in details that are not intended to considered. The ship is allowed instant change of direction with no loss of speed so that you don’t have to waste time factoring speed loss into the equation or the time taken to turn. The questioner does not want you to consider these things. Ship B is allowed to react instantly to Ship A because the questioner does not want you to consider reaction times. The questioner does not want you to come up with an answer such as “it depends on the manoeuvrability of the ships and the reactions of the captains.” This is quite reasonable. It is NOT reasonable to then conclude that this allows effectively no change in direction at all, that just entirely misses the point of the question.

In short, your interpretation of the question is silly and tortured. The question itself is fine as far as these type of questions go. This is not like the aeroplane on a treadmill where the question itself contained a fundamental flaw that required you to make additional assumptions.

How did you get through school physics? Did you pick apart all of the physics problems in the same way?

94

Two points (having no physical dimension) would never touch, two real world ships would as soon as they got close enough for the pivoting bow or stern to intersect.

95

The fact that the question regards ships makes it pointless to worry about crap like infinitely approaching collision but never getting there. There are manyr other factors like where in the boat the pivot position is, does the boat “skid” during a turn, what the relative lengths and widths of the ships are, etc. All of thos factor far outweigh the asymptotic approach of the ships.

If the OP wanted to be pedantic he could have done so using points, plains, lines, etc. To me it is an interesting question: can a ship with the same speed of another avoid capture by manouevering, or should it just maintain a steady course?

96

You make an unwarranted assumption. You assume you know what the poster of the OP intended. You do not.

I might well, as a teacher, offer a problem like this (I teach math, so it’s not so far off). I wouldn’t necessarily be interested in what a person decided was the “correct” answer; rather, I’d want to know the thought process used to answer it.

For someone who focused (as Xema has) on the fact that the problem as worded allows A to stay ahead of B, I would ask that they consider whether or not they are being overly pedantic in their interpretation of the problem. After all, in real life, one is often confronted with problems where one is required to interpret meaning from words that aren’t necessarily specific as to the underlying intent. Conversely, to a person who dismissed such an interpretation, I would ask that they consider the fact that they may be imposing their own viewpoint on the situation (much as I’ve said to sailor), and might want to clarify before making the assumption. And to a person who didn’t consider the interpretation at all, I’d ask if they considered other interpretations of the problem, to drive home the point that it’s not always a good thing to simply charge ahead without considering alternate points of view.

Here, the poster of the OP may well have been positing a problem designed to achieve what you and sailor have posited. But by the same token, the whole point to the question may be to see if someone spots the fact that the instantaneous changes can occur sequentially, without passage of time, thus allowing A to stay ahead of B; in short, leading you down the garden path of false assumption only to point and laugh. I doubt that’s what was intended, but the way the question is asked, one simply cannot dismiss it as an incorrect assumption.

Perhaps the better way to approach the “answer” for someone with sailor’s outlook would be to say: “Well, if the direction can change instantaneously, and thus A is allowed to instantaneously change the way the ship is pointed, then instantaneously (without passage of time) turn it back to the original heading, then B would never be able to catch up. But this seems trivially simple, so I’ll assume that this was not intended, and that the problem simply didn’t consider this possibility. I will add, then, the added stipulation that a change of direction must be followed by some finite amount of time before Boat A can again change direction. With this added condition, here is my answer…”
Yeah, I was an attorney at one point. Your reason for asking? :stuck_out_tongue:

97

On an infinite sea? Steady course all the way. Capture would be impossible.

98

This isn’t true. The optimum course course for B is to aim where A is going making an isosceles triangle between A, B, and the interception point where A-Interception point equals B-Interception point.

99

Could ship A could use ship B’s behavior to its advantage? Let’s imagine that ship B will always behave both optimally and instantaneously. Let’s also assume that there is a finite distance (time elapsed) a ship must travel after changing direction. (This should avoid the instantaneous-left-right-left-right as straight argument.) Now let’s imagine that the ships are on a Cartesian plane. Ship A is at (0,1) and ship B is at (0,0). The units describe the smallest finite time that the ships can operate in. That is, if ship A, which starts at (0,1) is headed up the Y-axis and doesn’t decide at that moment to change direction, it IS going to end up at (0,2).

So let’s say, for the sake of argument, that ship A decides to veer at a right angle, directly east, so that at the next unit of time it will end up at (1,1). Ship B instantaneously detects this, as we know, so where does it go? Well, what point will be closest to (1,1) after the elapsed unit of time? The 45-degree angle will take her to the closest point, so at the end of the unit of time she will be at (sqrt2/2,sqrt2/2). And of course, she will have gained a lot of ground. If we keep this up, it won’t be long at all till she catches ship A (asymptotically included, if you like). I mean, geometry is what it is, and the hypotenuse is far more efficient than the sum of the two legs.

But here’s the rub. Ship B can’t just maintain course after the unit of time has elapsed. It will have to change course itself. It will have to, independently and on its own, direct itself toward the point where ship A will be after two units of time as it continues on its direct course (y = 1).

So at the end of two units of time, there is a specific point where ship B will be—again, assuming it acts optimally and instantaneously. Can ship A use this knowledge to its advantage and direct itself to a point that after two units of time is actually farther away from ship A than it was in the beginning?

Of course it can. If we keep THIS up, ship A is going to be out of ship B’s sight before too long.

This is a long way of saying that ship B always wins when it has the benefit of reacting as opposed to acting, but that ship A can win quite easily if it can somehow depend on that reaction.

100

Well I think you can and should disregard any reading of the question that makes the question meaningless. To allow instantaneous changes of direction such that there is no measurable change in direction at all renders the requirement of Ship A to change direction redundant and so the question itself becomes meaningless. Therefore this interpretation of the question is pointless and of no value and should be disregarded. To not only seriously consider this interpretation but to also consider it to be as valid as, if not more so than, the more obvious interpretation seems to demonstrate a bizarre lack of common sense. To also quibble over the difference between “course” and “direction” just seems to show a willingness to over complicate things.