I. E. Irodov Solution 1.9

Solution to I. E. Irodov in General Physics and H. C. Verma in Concept of Physics is like a bible for student who are appearing for IIT-JEE, JEE/Main and JEE/Advance, UG NEET, AIIMS or any other Engineering and Medical entrance examination. All the questions in these books are of high level, which requires all basics to applied concept of physics. We here makes your task very easy, we have presented complete solution with detailed explanation step by step. The solution of these books teaches students also teachers in a suitable manner and then tests you with some tricky questions. To answer these questions you need to have thorough understanding of the concepts and this is where most students falter.

Problem: 1.9
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?

Solution: 1.9

Here the boats speed is less than that of the river, so there is no way that the boat can move in a direction perpendicular to the flow. Thus there will be a drift. Suppose that the boat travels at an angle \(\theta \) to the perpendicular direction as shown in the figure.
Further suppose that the boat speed is \(v\) and the river speed is \(u\). Then, the boat will travel at a speed \(v\cos \theta \) towards the bank. Thus, if the river is \(x\) units wide it will take \(\frac{x}{{v\cos \theta }}\) time for the boat to reach the other shore. During this entire time however, the boat is drifting with the river with a speed \(\left( {u - v\sin \theta } \right)\). Thus, the drift will be \(\frac{{x\left( {u - v\sin \theta } \right)}}{{v\cos \theta }}\)

The drift will be minimized when,
\(\frac{{d\left( {x\frac{{u - v\sin \theta }}{{v\cos \theta }}} \right)}}{{d\theta }} = 0\)
or, \(\frac{{ - v\left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right) + u\sin \theta }}{{{{\cos }^2}\theta }} = 0\)
or, \(\sin \theta = \frac{v}{u} = \frac{1}{n}\)
That the second derivative is positive at this value indicating that the value obtained in indeed a minima.