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 Post subject: More travel funPosted: Wed, 17 Aug 2011 03:37:07 UTC
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A[U@u].........................k..........................[V@v,W@w]B
Straight path AB, length k.
U (speed = u) is at A, V (speed = v) and W (speed = w) are at B.
All leave at same time.

U meets V at point C, picks up V, turns around, heads towards A, drops V
off at point D, turns around and heads towards B; V continues towards A.

U meets W at point E, picks up W, turns around, heads towards A:
they arrive at A at same time as V.

When V and W were "aboard" U, speed was u.
No loss of time at pick-ups and turn-arounds!

BC = a, AD = b and BE = c.

What is k in terms of a,b,c ?

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 Post subject: Re: More travel funPosted: Wed, 17 Aug 2011 06:19:25 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 7624
Location: NCTS/TPE, Taiwan
Denis wrote:
A[U@u].........................k..........................[V@v,W@w]B
Straight path AB, length k.
U (speed = u) is at A, V (speed = v) and W (speed = w) are at B.
All leave at same time.

U meets V at point C, picks up V, turns around, heads towards A, drops V
off at point D, turns around and heads towards B; V continues towards A.

U meets W at point E, picks up W, turns around, heads towards A:
they arrive at A at same time as V.

When V and W were "aboard" U, speed was u.
No loss of time at pick-ups and turn-arounds!

BC = a, AD = b and BE = c.

What is k in terms of a,b,c ?

Assuming D is between A and B?

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 Post subject: Re: More travel funPosted: Wed, 17 Aug 2011 09:42:16 UTC
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Location: Ottawa Ontario
outermeasure wrote:
Assuming D is between A and B?

Yes, of course...sorry...plus E also.

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 Post subject: Re: More travel funPosted: Wed, 17 Aug 2011 09:51:30 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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Location: NCTS/TPE, Taiwan
Spoiler:
From the start to when U,V meet at C, U travelled distance k-a at speed u and V travelled distance a at speed v.

From drop off at D to all meet at A, U travelled distance DE+EA=(k-b-c)+(k-c)=2k-b-2c at speed u and V travelled distance b at speed v.

Hence , which rearranges to .

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 Post subject: Re: More travel funPosted: Wed, 17 Aug 2011 20:32:30 UTC
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Location: Ottawa Ontario
Yep; matches mine; as example:
a = 40, b = 20 and c = 90 makes k = 120 ; (speeds u:v:w = 10:5:6)

If the given a was AC instead of BC (thus BC = k - a), then we have a slightly different:
a / (k - a) = (2k - b - 2c) / b , leading to a quite more complicated:
k = [SQRT(d^2 - 8e) + d] / 4 where d = 2a + b + 2c and e = 2ac

I'm unable to make that simpler; can you?

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