Are you really going to argue using the losing Zeno’s paradox position?
:rolleyes: I’m not sure whether this is meant as actual argument or just as snide comment. On the off chance that you are serious: No, and this really has nothing to do with Zeno’s paradox.
Let’s review. Zeno argues that we can’t get from X to Y, because to do that we have to perform an infinite number of operations. First we have to get halfway to Y, then we have to go half of the remaining distance, and so on. This argument fails because we can do each operation in half the time it took to perform the previous one. So the time series converges: 1+1/2+1/4+…=2. We can perform all of these operations in finite time, and kinematics is saved.
The situation Sage Rat and I are talking about, which I repeat really has nothing to do with Zeno’s paradox, has A running around in a circle at constant speed, while B, starting from the center of the circle and moving at the same speed, tries to catch A. I assert that the distance between A and B, for large time, is approximately 2/t. This means that if it takes time T for B to close half the distance to A, then it takes the same amount of time T for B to close half of the remaining distance to A, and so on. Notice that the time series no longer converges: 1+1+1+… is not finite. Contrast this to Zeno above. Do you see the difference?
It is different but it is still a similar concept. Eventually the ships will be so close that in the real world they will be called “touching”.
I believe this is wrong and my reasoning is very simple:
Both ships travel at the same speed. All B has to do is alvays point directly at A. If A is heading directly away from B then the distance remains the same. If B points in any other direction then the distance diminishes. Therefore if B is forced to change direction for any reason it follows A will catch up.
Only one direction maintains distance and all others diminish distance.
Having read the whole thread, I think the key point is Omphaloskeptic 's observation that the problem is underdetermined. So each of us may choose their own version to solve.
I choose to dance on the pin head of “instantaneous”.
Per the OP, Ship A is required to change course every so often. Since these changes are “instananeous”, they take zero time and therefore cover zero distance regardless of the ship’s speed.
There is nothing to prevent ship A from immediately performing another instananeous turn the other direction back to the original course. Which also covers zero distance. And therefore the actual path of ship A is indistinguishable from the path which had no course change at all.
Therefore, Ship A can manuever like this forever and ship B can do no better than hold its distance. That is A’s optimal strategy and completely negates any B strategy.
Now this *assumes *that A can conduct two zero-duration manuevers *with zero recovery interval *between them. If the recovery interval is constrained to be non-zero, then we get to the classical triangle or spiral pursuit cases.
An interesting case comes in when B’s reaction time is less than A’s (non-zero) recovery interval. i.e. A can change course no more often than every 1 minute, whereas B can only change course 2 minutes after observing a change in A’s course. Off top of head, it seems that A could induce B to open the distance by continuously forcing lag pursuit during the second minute of the reaction interval.
Which I take to mean B can react in whatever way he wants and for the purposes of this problem assume he reacts in the manner most convenient for him to catch A.
Not true. Sometimes true but not always true.
Always true if B is trying to catch A.
This argument does not hold water for me. A “change in course” that is infinitessimally small is, in fact, no change in course. It is like saying you rolled through a stop sign but for an infinitessimal moment your speed was zero. No. It was not zero. Similarly a ship cannot change course for an infinitessimal. If it changed course then this change in course is measurable and real. If there is no measurable change in course then there is no change in course. This is just playing with infinitessimals so that they simultaneously have and have not effects. You cannot do that.
And I am not even going to address Zeno’s paradox kind of objections.
It is a matter of vectors.
If A and B have the same speed and course the distance remains the same. But if either are different than the distances changes. Think about two fighters in a dog fight same thing can occure.
If A makes a 10 degree change to the right, to intercept B is also going to turn to the right but less than 10 degrees. Now you have a vector that is differeent. The larger the turn and the longer that it is held by A the smaller that vector becomes.
Now If B also makes a 10 degree change to the right at the same time, then no closing distance.
In the real world, perhaps, but not under the conditions of the OP, which specifies that “Direction changes are instant and involve no loss of speed”. This means that A can change direction, say, 10 degrees left and then 10 degrees right with no loss of speed or distance. If you claim otherwise, you are working on a different problem.
This is not possible if direction changes are instantaneous and speed is unaffected by them, both of which are specified in the OP. Distance can change only after some finite time.
This is incorrect. Consider the case where A is 2 miles north of B and both are headed due west at the same speed: their distance is not decreasing.
This suggests another scheme that meets the conditions of the OP and allows A never to be overtaken: A makes the required occasional “instant” direction changes, then instantly matches whatever direction B has chosen (we are assuming A has the same instant reaction capabilities as B). Under this scheme distance between the ships never decreases.
Except when defined as such during the statement of the problem.
To be precise, the OP speaks of changes in direction, not course. It quite specifically states that these “are instant and involve no loss of speed.”
For a real-world variant of this problem, we can (as Snnipe70E suggests) consider a WW-I style dogfight between two aircraft.
In that case direction changes certainly do consume time and cost distance (since pretty much any real-world vehicle incurs losses during a turn). There’s also a significant element of pilot reaction time, which is of course never instantaneous. Once of the strongest predictors of the eventual winner will be the maneuverability of the two aircraft: the one that can turn tighter (i.e. more degrees course change per second) has a big edge.
I disagree. This is just playing with infinitessimals in a way similar to Zeno’s paradox so I am not getting into it. The OP cannot redefine calculus.
You cannot have “change in course” defined in such a way that it does not produce the effect of a “change in course”. That is just playing with words. You cannot define “change in course” as “no change in course” and have a meaningful scenario.
This is like the question we had some time ago about a slanted line being composed of infinite infinitessimals of horizontal and vertical lines and arriving at a contradiction. No. If you want to play with infinitessimals you better know your calculus or you will arrive at apparent contradictions.
Instantaneous = Zero?
I need to look at my calculus book again…
This would be true if the OP used the words “change in course.” The OP did not. The OP used the words “must change direction.”
It helps, when arguing, to argue using the correct assumptions.
Nevertheless, I agree with your assertion that the “problem” is reduced to the absurd if we take what the OP wrote to mean that Ship A can engage in changes in direction that don’t result in changes in “course” or loss in relative velocity in the original direction. Perhaps the OP will be willing to clarify the posited problem given this issue.
Some of you guys seem intent on using the inherent fuzziness of the English language to try and make a simple question much more complicated than it needs to be. If a change of direction is instantly followed by another change of direction such that a straight line is described, then why would you even bother posing the problem? You may as well say, “two ships are traveling line astern at the same speed, does the rear ship catch the first one?” I think it’s quite clear that ship A is required to change direction periodically and that it must be a measurable change sustained for a period of time. I think it is also quite clear that seeing as ship B was specifically given the ability to immediately respond to A and ship A was given no reciprocal ability, that ship A does NOT have such an ability.
What is the difference for you between change in direction and change in course? Because to me, an engineer and a sailor, they are equivalent. Course is the actual direction of movement of a vessel. Just the nautical term for “direction”. Maybe you are thinking of “heading”.
We can start arguing about the many slightly different terms (course, course made good, course over ground, heading, etc.) but I do not believe they add clarity but rather that they will just serve to confuse the issue. I think the OP has a simple and clear interpretation and trying to come up with confusing and nitpicky issues does not help. Mostly those who are coming up with the nitpicky issues just do not understand calculus and are diverting the issue to peripheral things they think they understand.
Here is a scenario: At t=0 object A starts at coordinates (1,0) and B at (0,0), both with a speed of 1. A can move in any direction it wishes while B will always move directly towards A.
If A proceeds directly along the X axis with no change in direction then the distance between them is always 1. The instant A changes direction B starts gaining. I challenge anyone to come up with an actual function that describes the travel of A and which contradicts this (and, obviously, which is consistent with standard calculus; invented, imaginary alternate universes are not allowed).
Yup, it never fails. The OP is quite clear to me and I think it is worded pretty much like any problem in a physics class.
The OP never defines “change in course” - as previously noted, it mentions “direction changes”. And it quite clearly states that these have zero duration. From this it necessarily follows that they need have no effect on course.
I agree that what’s specified in the OP is not applicable in the real world. I disagree that this means that a clearly specified problem must necessarily be changed in order for it to be discussed.
Our different positions on this question could perhaps be summarized as follows:
Me: Under the impossible specifications of the OP, the answer to the question is X.
You: The OP specifies some things that are impossible - if we instead consider what’s possible, the answer to the question is Y.
Yes, the ships will asymptototically approach each other. The OP, however, distinguished between
Note that this is still different from the point of Zeno’s paradox, which is that an infinite number of events can happen in finite time. It’s more like the engineer-and-mathematician joke: “Close enough for all practical purposes.”
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][]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.