Problem 3 - Master Resource Book in JEE Main Physics

Question

A boat takes two hours to travel \(8 \mathrm{~km}\) and back in still water. If the velocity of water \(4 \mathrm{kmh}^{-1}\), the time taken for going ups tream \(8 \mathrm{~km}\) and coming back is (a) \(2 \underline{\mathrm{h}}\) (b) \(2 \mathrm{~h} 40 \mathrm{~min}\) (c) 1 h \(20 \mathrm{~min}\) (d) cannot be estimated with the information given

Step-by-Step Solution

Answer

The correct choice is (b) 2 hours 40 minutes.

Step 1: Understand Boat Speed in Still Water

The given exercise tells us that the boat can travel 8 km in still water in one direction and back within 2 hours. This allows us to compute its speed in still water. Since it covers a total of 16 km in 2 hours, the speed of the boat in still water is \( \frac{16 \text{ km}}{2 \text{ hr}} = 8 \text{ km/hr} \).

Step 2: Calculate Effective Speeds

With the velocity of water being 4 km/hr, we need to determine the effective speed of the boat when going upstream and downstream. - **Upstream (against the current)**: The effective speed is reduced by the flow of the water, giving **\( 8 \text{ km/hr} - 4 \text{ km/hr} = 4 \text{ km/hr} \)**.- **Downstream (with the current)**: The effective speed increases by the flow of the water, giving **\( 8 \text{ km/hr} + 4 \text{ km/hr} = 12 \text{ km/hr} \)**.

Step 3: Calculate Time for Upstream Travel

Using the speed for upstream travel, you need to find the time it takes to go 8 km upstream with the effective speed of 4 km/hr.The time taken for the upstream travel is \( \frac{8 \text{ km}}{4 \text{ km/hr}} = 2 \text{ hours} \).

Step 4: Calculate Time for Downstream Travel

For the return trip downstream, calculate the time to travel 8 km with an effective speed of 12 km/hr.The time taken for downstream travel is \( \frac{8 \text{ km}}{12 \text{ km/hr}} = \frac{2}{3} \text{ hr} \), which equals 40 minutes.

Step 5: Add Time for Complete Journey

You need to combine the time for both the upstream and downstream trips to determine the total time taken. - **Upstream**: 2 hours - **Downstream**: 2/3 hour (or 40 minutes) Total time = 2 hours + 40 minutes = 2 hours 40 minutes.

Step 6: Choose the Correct Option

From the options provided: (a) 2 hours (b) **2 hours 40 minutes** (c) 1 hour 20 minutes (d) cannot be estimated with the information given The correct choice is (b) 2 hours 40 minutes.

Key Concepts

Boat Speed Calculation

Understanding how a boat's speed is calculated in still water is crucial to solving riverboat problems. In still water, the boat's speed is the distance it can cover in a specific time without any interference from currents. For this exercise, the boat travels 16 km (8 km each way) in 2 hours. This calculation is straightforward:

• The speed is calculated by dividing the total distance by the total time: \( \frac{16 \text{ km}}{2 \text{ hr}} = 8 \text{ km/hr} \).

This calculated speed is only in still water. It's essential to distinguish this from situations where the water's current affects speed.

Upstream and Downstream Motion

Navigating a river involves moving both upstream and downstream, and understanding these motions is key to effectively solving related problems.

• Upstream: This refers to moving against the current. The water's flow reduces the boat's speed. For this exercise, the boat's effective speed upstream is found by subtracting the water's velocity from the boat's speed in still water: \( 8 \text{ km/hr} - 4 \text{ km/hr} = 4 \text{ km/hr} \).
• Downstream: Here, the boat moves with the current, increasing its speed. Thus, the boat's effective downstream speed becomes: \( 8 \text{ km/hr} + 4 \text{ km/hr} = 12 \text{ km/hr} \).

Always remember this adjustment in speed when assessing the boat's motion either upstream or downstream.

Effective Speed Calculation

Effective speed is the actual speed of the boat as it moves through water affected by the current. It varies based on the direction of travel relative to the current. For example:

• **Upstream Effective Speed:** This is the boat's speed in still water minus the current's speed, because the current works against the boat's movement.
• **Downstream Effective Speed:** This is the boat's speed in still water plus the current's speed, as the current aids in the boat's movement.

Such calculations help us find realistic travel times for boats moving in conditions influenced by river currents, crucial to planning and problem-solving in river navigation scenarios.

Time and Distance in River Currents

Calculating time and distance when a river's current is involved requires careful consideration of effective speed.For this problem, we calculate the time taken for both upstream and downstream movement separately:

• **Upstream Time:** With an effective speed of 4 km/hr, the time taken for 8 km is \( \frac{8 \text{ km}}{4 \text{ km/hr}} = 2 \text{ hours} \).
• **Downstream Time:** At an effective speed of 12 km/hr, the time taken for 8 km is \( \frac{8 \text{ km}}{12 \text{ km/hr}} = \frac{2}{3} \text{ hours} \), which equals approximately 40 minutes.
• **Total Journey Time:** Adding both the upstream and downstream times gives the complete journey time of 2 hours 40 minutes.

Understanding these calculations is crucial for estimating travel durations accurately, especially in environments influenced by natural elements.