When a boat travels in a river (strictly in a straight line), it can go either in the direction of flow of river (i.e downstream) or in the direction opposite the flow of river (i.e. upstrem ). Thus the boat's actual speed is more than by which it can move in stationary water while travelling downstram (as river's flow speed is added to it) and less while travelling upstream (as the boat moves against the flow of river).Based on the given information answer the following questions A boat going downstream in a following river overcome a raft at a point P. 1 h later it turned back and after some time passed the raft at a distance 6 km from point P.
Now, it instead of 6 km they have met at 8 km from point P. Find the speed of river

Options

A. 1 km/h
B. 2 km/h
C. 3 km/h
D. 4 km/h

Answer

To solve the problem, we need to analyze the motion of the boat and the raft in the river. Let's break it down step by step.

### Step 1: Define Variables
- Let the speed of the boat in still water be \( v \) km/h.
- Let the speed of the river (current) be \( u \) km/h.
- The speed of the raft is the same as the speed of the river, which is \( u \) km/h.

### Step 2: Analyze the Downstream Journey
When the boat travels downstream, its effective speed is:
\[ v + u \]
In 1 hour, the distance covered by the boat downstream is:
\[ \text{Distance}_{\text{boat}} = (v + u) \times 1 = v + u \text{ km} \]

The raft, moving downstream with the current, covers:
\[ \text{Distance}_{\text{raft}} = u \times 1 = u \text{ km} \]

### Step 3: Calculate the Distance Between the Boat and Raft
After 1 hour, the distance between the boat and the raft is:
\[ \text{Distance between them} = (v + u) - u = v \text{ km} \]

### Step 4: Analyze the Upstream Journey
After 1 hour, the boat turns back and travels upstream. Let \( t \) be the time taken for the boat to meet the raft again after it turns back.

The distance covered by the boat while going upstream is:
\[ \text{Distance}_{\text{boat}} = (v - u) \times t \]
The distance covered by the raft in the same time is:
\[ \text{Distance}_{\text{raft}} = u \times t \]

### Step 5: Set Up the Equation
The total distance covered by both the boat and the raft when they meet again is equal to the initial distance \( v \):
\[ (v - u) \cdot t + u \cdot t = v \]
This simplifies to:
\[ v \cdot t = v \quad \Rightarrow \quad t = 1 \text{ hour} \]

### Step 6: Total Time Calculation
The total time taken for the boat to meet the raft again is:
\[ \text{Total time} = 1 \text{ hour (downstream)} + 1 \text{ hour (upstream)} = 2 \text{ hours} \]

### Step 7: Distance from Point P
In the second scenario, the boat meets the raft at a distance of 8 km from point P. Thus, we can use the formula:
\[ \text{Speed of river} = \frac{\text{Distance}}{\text{Total time}} = \frac{8 \text{ km}}{2 \text{ hours}} = 4 \text{ km/h} \]

### Conclusion
The speed of the river is:
\[ \boxed{4 \text{ km/h}} \]