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.
The distance between the boat and raft at a later instant of time t is

Options

A. `(u+v)t`
B. `ut`
C. `vt`
D. `(u-v)t`

Explanation

To solve the problem, we need to determine the distance between the boat and the raft at a later time \( t \) after the boat overtakes the raft. Let's break down the solution step by step.

### Step 1: Understand the speeds involved
- The speed of the boat in still water is \( u \).
- The speed of the river current (and hence the speed of the raft) is \( v \).
- When the boat is moving downstream, its effective speed relative to the ground (or the bank) is \( u + v \) because it is moving with the current.

### Step 2: Determine the relative speed of the boat with respect to the raft
- Since the raft is floating with the river, its speed is \( v \).
- The relative speed of the boat with respect to the raft is given by:
\[ \text{Relative speed} = \text{Speed of the boat} - \text{Speed of the raft} = (u + v) - v = u \]

### Step 3: Calculate the distance traveled by the boat relative to the raft
- The distance between the boat and the raft after a time \( t \) can be determined using the formula:
\[ \text{Distance} = \text{Relative speed} \times \text{Time} \]
- Substituting the relative speed we found:
\[ \text{Distance} = u \times t \]

### Step 4: Write the final expression for the distance
- Therefore, the distance between the boat and the raft at a later instant of time \( t \) is:
\[ \text{Distance} = u \cdot t \]

### Final Answer
The distance between the boat and the raft at a later instant of time \( t \) is \( u \cdot t \).

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