A boat can be rowed in still water at a speed u. The boat is moving downstream in a river in which water flows at a speed v. There is raft floating in water and therefore moving along with water at speed v. Let the boat overtakes the raft at the moment t = 0.
Let the boat turns back at time `t=t_(0)` and starts moving towards the raft. The separation between the raft and boat after a time interval t' measured from the moment of turning back is

Options

A. `ut_(0)-ut'`
B. `ut_(0)-(u+v)t'`
C. `ut_(0)-vt'`
D. `ut_(0)-ut'+vt'`

Explanation

To solve the problem, we need to analyze the motion of the boat and the raft step by step.

### Step-by-Step Solution:

1. **Understanding the Initial Setup**:
- The boat can row in still water at a speed \( u \).
- The river flows at a speed \( v \).
- When the boat is moving downstream, its effective speed is \( u + v \) (boat speed + river speed).
- The raft moves with the river at speed \( v \).

2. **Distance Traveled by the Boat Before Turning Back**:
- The boat overtakes the raft at \( t = 0 \).
- Let the boat turn back at time \( t = t_0 \).
- The distance traveled by the boat downstream in time \( t_0 \) is given by:
\[ \text{Distance}_{\text{boat}} = (u + v) t_0 \]
- This distance is the initial separation between the boat and the raft when the boat turns back.

3. **Boat's Speed When Moving Upstream**:
- After turning back, the boat moves upstream against the river's current.
- The effective speed of the boat when moving upstream is:
\[ \text{Speed}_{\text{upstream}} = u - v \]

4. **Distance Traveled by the Boat and Raft After Turning Back**:
- Let \( t' \) be the time interval after the boat turns back.
- The distance traveled by the boat upstream during time \( t' \) is:
\[ \text{Distance}_{\text{boat}} = (u - v) t' \]
- The distance traveled by the raft downstream during the same time \( t' \) is:
\[ \text{Distance}_{\text{raft}} = v t' \]

5. **Calculating the Separation After Time \( t' \)**:
- The initial separation when the boat turns back is \( (u + v) t_0 \).
- After time \( t' \), the separation between the boat and the raft becomes:
\[ \text{Separation} = \text{Initial Separation} - \text{Distance}_{\text{raft}} + \text{Distance}_{\text{boat}} \]
- Substituting the distances:
\[ \text{Separation} = (u + v) t_0 - v t' + (u - v) t' \]
- Simplifying this expression:
\[ \text{Separation} = (u + v) t_0 - v t' + u t' - v t' \]
\[ = (u + v) t_0 - (v + v) t' + u t' \]
\[ = (u + v) t_0 - 2v t' + u t' \]
\[ = (u + v) t_0 - u t' \]

6. **Final Expression for Separation**:
- The final expression for the separation between the raft and the boat after time \( t' \) is:
\[ \text{Separation} = u t_0 - u t' \]

### Conclusion:
The correct answer is:
\[ \text{Separation} = u t_0 - u t' \]