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. We have two key velocities: the velocity of the boat relative to the river, denoted as \( v_{br} \), and the velocity of the river current, denoted as \( v_r \). Given that the river's velocity is greater than the boat's velocity, \( v_r > v_{br} \), we can derive expressions for both minimum drift and minimum displacement.

Understanding the Scenario

Imagine a river flowing from left to right. The boat is trying to cross the river at an angle \( x \) upstream. The boat's velocity relative to the water is \( v_{br} \), and the river's current is moving at \( v_r \). The angle \( x \) is measured from the normal (perpendicular) to the riverbank. The goal is to find out how far downstream the boat will drift while crossing the river and the total distance it travels.

Velocity Components

First, we need to break down the boat's velocity into components:

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

Time to Cross the River

The time \( t \) it takes for the boat to cross the river can be calculated using the width of the river \( d \) and the perpendicular component of the boat's velocity:

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

Calculating Minimum Drift

The drift \( D \) is the distance the boat is carried downstream by the river current during the time it takes to cross. This can be expressed as:

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

Substituting the expression for \( t \), we get:

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

Finding Minimum Displacement

The total displacement \( S \) of the boat can be found using the Pythagorean theorem. The boat travels a distance \( d \) across the river and drifts \( D \) downstream. Thus, the minimum displacement is:

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

Substituting for \( D \), we have:

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

Factoring out \( d^2 \) gives:

\( S = d \sqrt{1 + \left(\frac{v_r}{v_{br} \cos(x)}\right)^2}

Summary of Results

To summarize, we derived the following expressions:

• Minimum Drift: \( D = \frac{v_r \cdot d}{v_{br} \cos(x)} \)
• Minimum Displacement: \( S = d \sqrt{1 + \left(\frac{v_r}{v_{br} \cos(x)}\right)^2} \)

These equations help us understand how the boat's angle and the river's current affect its path across the river. The greater the river's velocity relative to the boat's velocity, the more significant the drift and displacement will be.