June 12, 2008

Q. 1.9 - "Problems in General Physics" by I.E. Irodov

$\textbf{Question}$:

A boat moves relative to water with a velocity which is $n$= 2.0 times less than the river flow velocity. At what angle to the stream direction must the boat move to minimize drifting?

$\textbf{Answer}$:

Drift is the distance that the boat is carried downstream by the river while it tries to cross it.

Image diag

From the figure above, the boat goes across the river with speed $(v/n) \cos\phi$and it is carried downstream with a speed $(v-(v/n) \sin\phi)$. Assuming that the width of river is $w$, time taken by the river to cross the river is $\frac{w}{(v/n) \cos\phi}$.

In this time the distance by which the boat is carried downstream is $(v-v\sin\phi/n) \times \frac{w}{v \cos \phi/n}$=$ w\frac{n-\sin\phi}{\cos\phi} $

This drift will be minimum when:

\begin{displaymath}\frac{d(w\frac{n-\sin\phi}{\cos\phi})}{d\phi} = 0\end{displaymath}

\begin{displaymath}\Rightarrow \frac{d(n\sec\phi - \tan\phi)}{d\phi} = 0\end{displaymath}

\begin{displaymath}\Rightarrow n\sec\phi\tan\phi - \sec^2\phi = 0\end{displaymath}

\begin{displaymath}\Rightarrow \sec\phi (n\tan\phi - \sec\phi) = 0 \end{displaymath}

\begin{displaymath}\Rightarrow n\sin\phi - 1 = 0\end{displaymath}

\begin{displaymath}\Rightarrow \phi = \sin^{-1}(\frac{1}{n}) \end{displaymath}

Using the specified value $\phi = \sin^{-1}(1/2) = 30^o$. Or $30^o + \frac{\pi}{2} = 120^o$w.r.t to the stream velocity.

Posted by Apurva Sharan at 1:23 PM
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2 comments:

I love Physics said...

Can you please clarify following

1. The ϕangle you have in your solution is an angle w.r.t to the vertical (as per the diagram). So is the answer correct?
2. Why have you not considered the relationship between the different given velocities i,e V bw = V b-V w, where V bw is velocity of boat w.r.t to water, V bis velocity of boat w.r.t to earth and V wis velocity of water w.r.t to earth.

June 27, 2008 at 8:22 PM
Apurva Sharan said...

1. Yes. The answer is correct. Do let me know what aspect is not clear to you.

2. This is already considered in the answer. Again, do let me know which aspect is not clear to you.

July 23, 2008 at 12:18 PM

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