Boat velocity wrt river = v _br

river velocity = v _r

a boat is moving across the river by making an angle ‘ x ‘ with the normal upstream. Derive expressions for minimum drift & minimum

displacement ,if, v _r > v _br

To tackle the problem of a boat moving across a river while making an angle with the normal, we need to analyze the velocities involved and how they interact. Let's break this down step by step to derive expressions for minimum drift and minimum displacement when the river's velocity is greater than the boat's velocity across the river.

Understanding the Scenario

We have a river flowing with a velocity \( v_r \) and a boat that has a velocity \( v_{br} \) relative to the water. The boat is moving at an angle \( x \) upstream with respect to the normal (perpendicular) direction to the riverbank. The goal is to find the conditions under which the boat experiences minimum drift downstream while crossing the river.

Velocity Components

First, let's break down the velocities into components:

• The component of the boat's velocity across the river (perpendicular to the flow) is \( v_{br} \cos(x) \).
• The component of the boat's velocity upstream (against the river flow) is \( v_{br} \sin(x) \).

Drift Calculation

The drift of the boat is caused by the river's current. The time \( t \) it takes for the boat to cross the river can be calculated using the width of the river \( d \) and the component of the boat's velocity across the river:

\( t = \frac{d}{v_{br} \cos(x)}

During this time, the river carries the boat downstream. The drift \( D \) can be expressed as:

\( D = v_r \cdot t = v_r \cdot \frac{d}{v_{br} \cos(x)}

Minimum Drift Condition

To minimize the drift, we need to maximize the cosine component, which occurs when \( x \) is minimized. The minimum drift occurs when the boat is aimed directly across the river (i.e., \( x = 0 \)). In this case, the drift simplifies to:

\( D_{min} = \frac{v_r \cdot d}{v_{br}}

Displacement Analysis

Next, let's consider the total displacement of the boat from its starting point to its landing point on the opposite bank. The total displacement \( S \) can be calculated using the Pythagorean theorem, considering both the distance traveled across the river and the drift:

\( S = \sqrt{d^2 + D^2}

Substituting the expression for drift, we get:

\( S = \sqrt{d^2 + \left(\frac{v_r \cdot d}{v_{br}}\right)^2}

Minimum Displacement Condition

To find the minimum displacement, we can differentiate \( S \) with respect to \( x \) and set the derivative to zero. However, intuitively, we can see that the minimum displacement occurs when the boat is aimed directly across the river, which again corresponds to \( x = 0 \). Thus, the minimum displacement simplifies to:

\( S_{min} = \sqrt{d^2 + \left(\frac{v_r \cdot d}{v_{br}}\right)^2}

Final Expressions

In summary, when the river's velocity is greater than the boat's velocity across the river, the expressions for minimum drift and minimum displacement are:

• Minimum Drift: \( D_{min} = \frac{v_r \cdot d}{v_{br}} \)
• Minimum Displacement: \( S_{min} = \sqrt{d^2 + \left(\frac{v_r \cdot d}{v_{br}}\right)^2} \)

These expressions highlight the relationship between the velocities of the boat and the river, as well as the angle at which the boat is aimed. Understanding these dynamics is crucial for navigating effectively in flowing water.