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.
Find the speed of river

Options

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

Answer

To solve the problem, we will break it down step by step.

### Step 1: Define Variables
Let:
- \( U \) = speed of the river (in km/h)
- \( v \) = speed of the boat in still water (in km/h)

### Step 2: Analyze the Boat's Journey Downstream
The boat travels downstream for 1 hour. The distance covered by the boat downstream in this time is:
\[ \text{Distance}_{\text{boat}} = (v + U) \times 1 = v + U \text{ km} \]
The raft, which is floating with the river, travels downstream for 1 hour as well:
\[ \text{Distance}_{\text{raft}} = U \times 1 = U \text{ km} \]

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

### Step 4: Boat Turns Back and Meets the Raft
After 1 hour, the boat turns back and starts moving upstream. The raft continues to float downstream. Let \( t \) be the time taken for the boat to meet the raft after it turns back.

During this time \( t \):
- The distance traveled by the boat upstream is:
\[ \text{Distance}_{\text{boat}} = (v - U) \times t \]
- The distance traveled by the raft downstream is:
\[ \text{Distance}_{\text{raft}} = U \times t \]

### Step 5: Set Up the Equation for Meeting Point
The total distance covered by both the boat and the raft must equal the initial distance between them:
\[ (v - U) \times t + U \times t = v \]
This simplifies to:
\[ vt - Ut + Ut = v \]
Thus:
\[ vt = v \]
From this, we can conclude that:
\[ t = 1 \text{ hour} \]

### Step 6: Total Time and Distance
The total time taken from the start until they meet again is:
\[ \text{Total time} = 1 \text{ hour (downstream)} + 1 \text{ hour (upstream)} = 2 \text{ hours} \]
The distance between point P and point R where they meet is given as 6 km.

### Step 7: Calculate the Speed of the River
Using the distance and total time, we can find the speed of the river:
\[ \text{Speed of the river} = \frac{\text{Distance}}{\text{Time}} = \frac{6 \text{ km}}{2 \text{ hours}} = 3 \text{ km/h} \]

### Conclusion
Thus, the speed of the river \( U \) is:
\[ \boxed{3 \text{ km/h}} \]