Problem 83 - Physics for Scientists and Engineers with Modern Physics

Question

The speed of a boat in still water is \(v .\) The boat is to make a round trip in a river whose current travels at speed \(u\). Derive a formula for the time needed to make a round trip of total distance \(D\) if the boat makes the round trip by moving (a) upstream and back downstream, and ( \(b\) ) directly across the river and back. We must assume \(u

Step-by-Step Solution

Answer

(a) \( T_1 = \frac{Dv}{v^2-u^2} \); (b) \( T_2 = \frac{D}{\sqrt{v^2-u^2}} \). Assumption: \( u < v \) ensures progress against the current.

Step 1: Understanding the Problem

We have to find the time needed for a boat to make a round trip in a river. The round trip can be either upstream then downstream or directly across the river and back. The boat's speed in still water is \( v \), and the river's current speed is \( u \). The total distance for the round trip is \( D \). We also assume that \( u < v \).

Step 2: Calculating Time for Upstream and Downstream Trip

To calculate the time for the round trip by moving upstream and downstream, consider that when moving upstream, the effective speed of the boat is \( v - u \), and when moving downstream, it is \( v + u \).- The time taken to go \( D/2 \) upstream is given by: \[ t_{up} = \frac{D/2}{v-u} \]- The time taken to go \( D/2 \) downstream is given by: \[ t_{down} = \frac{D/2}{v+u} \]Thus, the total time \( T_1 \) is:\[ T_1 = t_{up} + t_{down} = \frac{D/2}{v-u} + \frac{D/2}{v+u} \]

Step 3: Simplifying Total Time for Upstream and Downstream

Combine the expressions for upstream and downstream time:- First find a common denominator: \[ T_1 = \frac{D(v+u) + D(v-u)}{2(v-u)(v+u)} \]- Simplify the expression: \[ T_1 = \frac{Dv}{v^2-u^2} \]This is the time needed for a round trip when traveling upstream and downstream.

Step 4: Calculating Time for Directly Across the River and Back

When the boat moves directly across the river and back, the cross-stream speeds are orthogonal. However, the effective path due to the river's current forms the hypotenuse of a right triangle, where the boat travels at speed \( \sqrt{v^2 - u^2} \) to counter the current to cross directly.- The total distance to cross and back is \( D \), so the time taken \( T_2 \) is:\[ T_2 = \frac{D}{\sqrt{v^2-u^2}} \]

Step 5: Explaining the Assumption u < v

The assumption \( u < v \) is necessary because if the river's current speed was equal to or greater than the boat's speed in still water, the boat would not be able to make progress against the current, making upstream travel impossible.

Key Concepts

Boat Speed Calculation

Understanding the concept of boat speed in different water conditions is crucial for solving river navigation problems. When a boat moves in still water, its speed is represented by the variable \( v \). However, when in a river, the current affects this speed. To calculate effective boat speed, we differ between upstream and downstream movement.

• Upstream: The boat moves against the current; therefore, its effective speed is reduced by the current speed \( u \). The formula here is simply \( v - u \).
• Downstream: The boat moves with the current, increasing its effective speed. Hence, the formula becomes \( v + u \).

This understanding allows accurate calculation of the boat's performance in mixed water conditions.

River Current Effect

The river current naturally influences how a boat travels either upstream or downstream. In terms of physics problem solving, understanding this effect is essential for predicting boat speed and travel time. The strength of the current, denoted by \( u \), either hinders or aids the movement of the boat depending on direction.

• Upstream Effect: The current creates resistance, effectively slowing down the boat's progress as it works against the natural flow of the river.
• Downstream Effect: The current provides additional force, speeding up the boat's journey by pushing it along the direction of flow.

Recognizing these effects helps us understand why trips against the current are longer, influencing how we calculate travel times related to river currents.

Round Trip Time Formula

Calculating the time for a complete round trip in a river involves different formulas depending on the direction relative to the current.

• Upstream and Downstream: When a boat travels in both directions, we consider the effective speeds mentioned before. The time taken is the sum of: - Upstream time: \( t_{up} = \frac{D/2}{v-u} \) - Downstream time: \( t_{down} = \frac{D/2}{v+u} \) Combined, the total time \( T_1 \) is: \[ T_1 = \frac{Dv}{v^2-u^2} \]
• Directly Across and Back: For a path directly across the river and returning, the boat moves perpendicular to the current, forming a right triangle. Here, the total travel time \( T_2 \) simplifies to: \[ T_2 = \frac{D}{\sqrt{v^2-u^2}} \]

Calculating these separately shows how currents and directionality influence travel time.