Re: Solution to a problem!!

[rgrandy@ruf.rice.edu (Richard Grandy)]

I am a philosopher, not a physicist, so here is a qualitative argument to
an alleged solution.

Since J sprints faster than J rows, J wants to minimize rowing time, and
thus should row directly toward the opposite shore.  Since the river is 1
km wide and rowing speed in still water is 6km/hr it will take 10 minutes
to reach the opposite shore.  With the current of 4 km/hr, J will be  2/3
km downstream from the supposed beloved to be.  Sprinting along the shore
2/3 km at 10 km/hr will take 4 minutes.  So the total time for this
approach is 14 minutes.  Can you beat this with calculus or approximations?

Richard Grandy
Philosophy
Rice University

> The problem is quoted at the end.
> Assuming the river flows from left to right the boy must aim the
> boat at a point A located x km to the left from the girl. To minimize
> time he would plan to land at a point B on the right of the girl (not
> exactly at her location because he runs faster than he rows). The rest
> is just to compose the expression for the total time t=t1+t2 (rowing
> and running) in terms of the independent variable x (and other
> given parameters). Calculate the derivative of t with respect to x and
> equate it to zero. This will give you the best possible x. And the angle
> D you want is given by tan(D)=x/w, where w is the width of the river.
>
> To solve the problem without calculus (as far as I can tell) would
> be to use the "brute numerical force".  Calculate t for several x
> numerically, plot the curve and locate its minimum.
>                                                    Ludwik Kowalski
> James Harris wrote:
>
> > Dear Physics People,
> >
> > I gave the following problem as an extra credit exercise on a test for my
> > Honors physics.
> > What would you say the solution was? I believe it is from Tipler Physics. I
> > don't have an answer key for the book so I am not sure. A couple of kids got
> > into a pretty good debate (until the soccer coach arrived and chewed them
> > out) over what the solution is or was.
> > Jim
> > jharris@monad.net or jharris@newpisgah.keene.edu
> > Teacher: Monadnock Regional High School
> > Adjunct Faculty: Keene State College, Chemistry Department
> >
> > EXTRA CREDIT   Jack/Jill is strolling along the bank of a river 1 km wide
> > when the most beautiful/handsome girl/boy he/she has ever seen materializes
> > on the shore directly opposite him/her (perhaps she/he was beamed down?).
> > Fearing that she/he will disappear before he/she has a chance to establish
> > face-to-face communication, he/she quickly devises a plan to reach the
> > opposite shore in the shortest possible time. In the wildest of
> > coincidences, there is a rowboat beached on the shore right in front of
> > him/her. Jack/Jill is ecstatic, because it so happens that he/she's an
> > expert oarsperson. (He/she rows on the crew for an ivy league school.)
> > He/she knows that he/she can row at a speed of 6 km/h in still water, and
> > he/she estimates--as he/she sprints for the boat at 10 km/h--that the river
> > current has a speed of 4 km/h. Now, besides being an athlete and an
> > excellent judge of river velocities, Jack/Jill is also an accomplished
> > physics student. During his/her sprint to the boat, he/she computes the path
> > he/she must take from his/her side of the river to reach the girl/boy on the
> > opposite side in the shortest possible time. In general, his/her path
> > includes a diagonal trip across the river followed by a sprint along the
> > opposite shore to reach his/her goal. (Note that Jack/Jill has a standard
> > sprinting speed of 10 km/h.) Assuming that Jack/Jill did the physics
> > correctly, in what direction did he/she head the boat and how long did it
> > take him/her to establish first contact?