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Re: Solution to a problem!!

Je 10:47 PM 27.10.98 -0600, Richard Grandy skribis:

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

You have chosen what seems to me to be one of two "extreme" cases:

1. J always rows directly toward the other shore, and as you point out will
be swept downstream a distance 2/3 km. This gives your total time of 14
minutes.

2. J angles his approach to counteract the current, so that he actually
does travel directly across the river. So although his speed with respect
to the water is 6 km/hr, his speed with respect to his beloved is only 2
sqrt(5) km/hr. [From the right angle triangle with a hypotenuse of 6
km/hr, and one side of 4 km/hr, we get that the other side must be sqrt(6^2
- 4^2) =3D sqrt(36-16) =3D sqrt(20) =3D 2 sqrt(5) km/hr.]

In this case his approach time is 1 km / (2 sqrt(5) km/hr) =3D (1/10)sqrt(5)
hrs, approximately 0.22361 hrs or 13 min 25 sec.

So the second extreme case results in a slight time savings, even though J
can sprint faster than he can row.

However there is nothing to indicate that either of these is necessarily
optimal: perhaps J could steer at some angle which partially counteracts
the current and thus reduces his slippage downstream and his sprinting
time, while not decreasing "too much" the component of his velocity
directed across the river. And that's the optimization problem.

Ludwik Kowalski wrote this solution, which appears to me to be correct
except for one detail:

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.=20

Rather than aiming the boat at a point A located x km to the left of the
girl, I would think that he should aim at some fixed angle D (to be
determined as you describe below), such that he _arrives_ at a point
located x km to one side of the girl. [But perhaps I've misunderstood what
you meant.] If he were to actually head directly toward the point A his
path would be part of a spiral rather than a straight line, right? In
fact, if he always heads towards point A he will eventually reach it, since
his rowing speed is faster than the river speed. [I'd have to doodle a bit
to get the equation of the path. Something to do in my next committee
meeting, not now at 2:15 a.m.]

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=3Dt1+t2 (rowing
and running) in terms of the independent variable x (and other
given parameters).=20

Incidentally, it isn't possible to solve for t as a function of x, but no
matter, we can get a relation and differentiate implicitly.

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)=3Dx/w, where w is the width of the river.

Yes. Following this method, with the change mentioned above, I get:

x ~ 0.2635 km, t ~ 12 min, 39 sec. (t1 ~ 11.1 min, t2 ~ 1.6 min)

I found it easier to solve for D than x, but was still unable to get it in
closed form, but only as the solution of the equation

(-7/3) sec(D) =3D=3D cot(D) + tan(D)

Turns out the only solution in the range -pi/2 < D < pi/2 is=20

D ~ -0.4429110440736389 ~ -25.4 degrees

(Can be verified by graphing the lhs & rhs of the equation: 1=
intersection.)

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

Yes. =20

One final point: We may have found a maximum, or inflection point rather
than a minimum time. But comparing 12min 39s to our previous candidates of
14min and 13min 25s confirms that we're talking about a minimum.

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?

a. 25.4 degrees "upriver of straight across".
b. 12 min, 39 sec. =20

Thanks for a fun problem!

Ken

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