Motion inside and outside the Unit Boundary

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bperet
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Motion inside and outside the Unit Boundary

Post by bperet » Thu Oct 13, 2016 11:38 am

Larson's reasoning for determining speed in the time region is somewhat flawed. The logic he uses is that when you move "inside" the unit space boundary, space is fixed at 1 unit and only time progresses. He then concludes that "s" is replaced by "1/t" so speed, s/t becomes (1/t)/t = 1/t2. But "s" isn't replaced--when you are within the unit space boundary, s=1 and because it is at unit speed, he ignores it (just as we ignore "t" in the denominator of speed to get distance... s/1t = s).

The description in RS2 is different, because RS2 recognizes the "geometric reciprocals" of linear and angular velocities. Outside the unit space boundary, you have linear motion from 1->infinity. When you move inside the boundary, geometry inverts to angular motion from 1->0. Linear velocity is expressed mathematically as v1 and angular velocity is orbital, v2. Since s=1 inside the boundary, you end up with 12/t2 = 1/t2, the same as Larson.

The origin of the geometric reciprocal is that in 3 dimensions, lines and planes are "duals." Any (x,y) pair can represent either a line (0,0)->(x,y) or the corners of a plane (0,0)--(x,y). Lines are 1-dimensional structures, planes are 2-dimensional, so that gives the 1st to 2nd power relationship.
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Horace
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Re: Motion inside and outside the Unit Boundary

Post by Horace » Fri Oct 14, 2016 9:04 am

bperet wrote:Larson's reasoning for determining speed in the time region is somewhat flawed. The logic he uses is that when you move "inside" the unit space boundary, space is fixed at 1 unit and only time progresses.
It always puzzled me how one motion can affect and freeze another motion at 1u of space ? Is coincidence a prerequisite ?

I am mentioning two motions, because the movement "inside the unit space boundary" first requires an approach from afar to one unit of space, between two motions.
To be able to abstract space and talk about distances in space like that, we must have a relation between at least two motions with an assumption that time progresses unidirectionally wrt one.

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