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Topic: Snails linear motion (Read 3227 times) |
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chetangarg
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Snails linear motion
« on: Jul 5th, 2011, 9:47am » |
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Four snails travel in uniform, rectilinear motion on a very large plane surface. The directions of their paths are random,( but not parallel, i.e., any two snails could meet), but no more than two snail paths can cross at any one point. Five of the six possible encounters have already occured. Can we state with certainity that the sixth encounter will also occur.
Source : 200 puzzling physics problems
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towr
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Re: Snails linear motion
« Reply #1 on: Jul 5th, 2011, 9:52am » |
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If no two lines are parallel, then they all cross either in the past, present or the future. So it depends on how long ago the snails started their paths. If the 6th crossing is "further in the past" then when the snails started then it will not occur in the future.
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chetangarg
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Re: Snails linear motion
« Reply #2 on: Jul 5th, 2011, 10:04am » |
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Hi Towr i didn't understand your post. If we consider only straight lines then they will intersect but here time is also the criteria as the two snails should be on the intersection point of the lines at the same time
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rmsgrey
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Re: Snails linear motion
« Reply #4 on: Jul 6th, 2011, 7:48am » |
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If two objects both moving with constant velocity are on a collision course, then the bearing from one to the other is also constant. Contrariwise, if two objects are moving with constant velocity, the bearing from one to the other is constant, and the distance between them is decreasing, then they are on a collision course.
Additionally, for two objects moving with constant velocity, if the bearing from one to the other (mod 180 degrees) changes then it will continue changing in the same sense - in other words, unless its constant, it never returns to an earlier direction.
Take the two who have not encountered each other, call them A and B. A has encountered both C and D so they must both have constant bearing (though the sign of the range flips at the point of encounter) from A. Since C and D have encountered each other, they must have the same bearing from A as each other. Since they have both encountered B at different times, the bearing from A to B must also have been that same bearing at both times, so must be constant.
If the distance between A and B is decreasing, then they will encounter each other. If it's increasing, then they would have encountered if time extended further back.
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Grimbal
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Re: Snails linear motion
« Reply #5 on: Jul 8th, 2011, 1:23am » |
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I think the simplest is to consider the paths in space-time. It is a 3d space. 2 for location and 1 for time. A path is a straight line in space-time.
C and D have encountered, the 2 paths define a plane.
A has met both C and D at different times, its path must be within the plane defined by C and D.
The same applies to B, so A and B are within that same plane. If the paths are not parallel, they must cross somewhere in space-time. That measn they were at the same place at the same time. If it was not in the past, it must be in the future.
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