I'm working my way through some multivariable calculus using the 4th edition of the Hughes-Hallett text Single and Multivariable Calculus.
I'm having trouble with one of the questions – I cannot figure out how the answer the book gives is obtained. I suspect I'm thinking about the mechanics of the problem incorrectly. The book seems to assume some elementary knowledge of physics that I do not have.
Here is the problem:
A man wishes to row the shortest possible distance from north to south across a river which is flowing at 4 km/hr from the east. He can row at 5 km/hr. If there is a wind of 10 km/hr from the southwest, in which direction should he steer to try and go directly across the river? What happens?
The problem is in the review section of the first chapter on Vectors, and the chapter gives minimal information about forces.
If I let $\vec{w}$ be the velocity vector for the wind, then I have $\vec{w} = 5\sqrt2\ \vec{i}+5\sqrt2\ \vec{j}$. If I let $\vec{c}$ be the velocity vector for the current, I have $\vec{c}=-4\ \vec{i}$
My idea for a solution was to add $\vec{w}+\vec{c}$ to get the combined velocity vector of the current and wind. This would be the direction the boat would be carried if the man was not steering at all. Then I would have to find a new vector $\vec{v}$ so that $\vec{v}+\vec{w}+\vec{c}=-\vec{j}$, or perhaps some scalar multiple of j? I'm really not sure. The book doesn't expand much on how to deal with the magnitudes.
In any event, this method doesn't not give me the answer the book has. I get $\vec{v}=-3.07\vec{i}+-8.071\vec{j}$, which gives that the angle the man should steer is 20.8 degree west of south. But the book answer is 37.9 degrees west of south. The book also says if he steers in this direction, the boat will not move.
Any help or thoughts on this problem would be helpful. I've been thinking about it for so long – it seems like it should be simple so I can't figure out what I'm missing.
Best Answer
I think you have the forces of the stream and the wind correct (although it's hard to believe they want you to give them equal weight, but whatever ...) The problem is with the rowing.
The rower will row with some velocity $a\vec{i}+b\vec{j}$ He wants to row straight across the stream, so $v_i + w_i + a_i = 0$.
We still need a condition on $b$. We are told that he can row 5 km/hr. Therefore
$a^2 + b^2 = 25$.
With those two equations, you should be able to solve the problem.