Boat race
 

The boat has a constant speed relative to the water.
It is navigating (downstream) a flowing bendy river, for which the water velocity at the surface is known precisely at every point.

How do you work out the course to steer to get from start
to finish line in the shortest time?

David

[QUOTE=davieddy;256717]The boat has a constant speed relative to the water.
It is navigating (downstream) a flowing bendy river, for which the water velocity at the surface is known precisely at every point.

How do you work out the course to steer to get from start
to finish line in the shortest time?

David[/QUOTE]

2 possibilities I see depend on what you say relative means.

1)do nothing no matter what you do you go at a constant speed,
2)2 make sure to take the velocity of the river ( helping you) is the fastest for the area you are in because that should give you the fastest time (assuming equal distance).

of course if distance does change dramatically to follow the course of fastest water if may not always give the fastest time depending on how much farther you have to travel:

v=d/t so v/d=t

traveling 70kmph over 19[TEX]\frac{4}{9}[/TEX] m = 1 sec = 40kmph / 11[TEX]\frac{1}{9}[/TEX]

This one reminds me of an old riddle.

Why is the university boat race so hard?

[spoiler]Because there's 16 oars, 2 cox and only 20 minutes to do it in.[/spoiler]

Paul

I'm Number One and Not Afraid to Show It
 

[QUOTE=davieddy;256717]The boat has a constant speed relative to the water.
It is navigating (downstream) a flowing bendy river, for which the water velocity at the surface is known precisely at every point.

How do you work out the course to steer to get from start
to finish line in the shortest time?

David[/QUOTE]

If I understand this correctly, the boat has the same velocity as the water in which it is located. The boat could probably be considered as a point moving downstream, faced with choosing its next position. If all immediate points are available, the next point to follow is the point that is slightly faster than the present point. Depending on the number of bends to navigate before the finish line, staying centred in the river probably guarantees the fastest velocity until the finish line is in sight. At this point, a quick study must be made to follow the shortest path with the greatest velocity.

I think that this boat will win the race !

[QUOTE=davieddy;256717]The boat has a constant speed relative to the water.
It is navigating (downstream) a flowing bendy river, for which the water velocity at the surface is known precisely at every point.

How do you work out the course to steer to get from start
to finish line in the shortest time?

David[/QUOTE]

[url]http://en.wikipedia.org/wiki/Calculus_of_variations[/url]

[QUOTE=davieddy;256717]The boat has a constant speed relative to the water.[/QUOTE]

[QUOTE=9021951;276690]If I understand this correctly, the boat has the same velocity as the water in which it is located.[/QUOTE]

That's a trivial special case of the problem davieddy posed, namely, when the constant speed relative to the water is exactly zero. It is trivial because, no matter how you steered the boat, it would travel along the same path and take the same time to do so.

[QUOTE=davieddy;256717]How do you work out the course to steer to get from start
to finish line in the shortest time?[/QUOTE]

Are there any restrictions upon the steering. For example, must the heading vary continuously? differentiably? Is there a limit on how fast the heading can change, i.e., upon its first derivative. Same question in respect of higher derivatives?

Jokes out of the way before any "remedial physics"
 

[QUOTE=9021951;276690]If I understand this correctly, [/QUOTE]
You don't, but thanks for reviving the puzzle anyway.
I thought this would be taken as intended as a concise example
of a general type of problem that, given a precise model, computers
could eat up these days.


[QUOTE=xilman;257657]This one reminds me of an old riddle.

Why is the university boat race so hard?

[spoiler]Because there's 16 oars, 2 cox and only 20 minutes to do it in.[/spoiler]

Paul[/QUOTE]

Yep. Boat Race means one of two things this side of the pond,
but I made no mention of uni, (wh)ores or cocks.

However, while teaching physics at St Paul's school (located at the apex
of the Hammersmith bend) I coached rowing for 13 years, having rowed
for Exeter and St Catharine's 1st VIIIs.

David

PS some of you may need to check out "Cockney rhyming slang".

[QUOTE=Mr. P-1;276736]Are there any restrictions upon the steering. For example, must the heading vary continuously? differentiably? Is there a limit on how fast the heading can change, i.e., upon its first derivative. Same question in respect of higher derivatives?[/QUOTE]

Either you ar masquerading as RDS, or you have never been in a car,
plane, boat or walked in a straight line in your life:smile:

David

PS Forget about quantum leaps.
Planck/Bohr/Feynman/Sinclair and above all Einstein would tell you
"If you think you understand this, YOU DON'T"

[QUOTE=davieddy;276771]Either you ar masquerading as RDS, or you have never been in a car, plane, boat or walked in a straight line in your life[/QUOTE]

Of course not. I travel everywhere by unicycle.

[QUOTE=Mr. P-1;276799]Of course not. I travel everywhere by unicycle.[/QUOTE]
Isn't that a tadge one-dimensional?

David