Re: Solution to a problem!!

[Ludwik Kowalski <kowalskiL@Mail.Montclair.edu>]

Robert Blumenthal wrote:

> All the proposed solutions I have seen seem to assume that the
> solution path is a straight line. You aim the boat and maintain
> your heading. This may be so but it needs proving

LET ME TRY.

1) First simplify the problem a little by assuming that the
speed of rowing is the same as the speed of running. In this
case the solution is trivial, the boat should arrive to the other
shore exactly where the girl is. The tangent of the desired
angle must be equal to the speed of water divided by the
speed of rowing.

We can now ask your question. How do we know that the
strategy of keeping the same angle is the best? Because any
other strategy would yield a curved line between the two
fixed points in space. The straight line is the shortest, in
classical geometry.

2) Our problem can be analyzed in a similar way. The rowing
velocity (in the shore's frame of reference) for any chosen
"fixed angle" is a straight line. We already know where
the boy should reach the shore to yield the shortest time
(for the fixed angle strategy). Then he runs toward the girl
along the shore. This defines a right triangle ABC, where A
is the starting point for the boat, B is the landing point, and
C is where the girl is waiting. (Please make a drawing to
follow my logic.)

The reduction of time (from what we have already calculated)
is impossible if the boat turns to the right from AB because
both rowing and running become longer. But suppose it turns
left and lands at a point D which is closer to C than B (keep
drawing).  Can the overall time be shorten in this way? Not
at all. Let me prove this by "the reduction to absurdum".

Suppose that the overall ADC time becomes shorter than ABC.
Note than AD is now a curved line. Then the new time T' must
be compared with the ADC time T" for which AD is a straight
segment. Clearly T" must be shorter than T' (look at your
drawing). But this would contradict our previous finding
according to which the ABC time is shorter than the time
along the sides of any other right triangle.

Is this a convincing argument?
Ludwik Kowalski