An observer on ground sees a boat cross a river of width 800 m perpendicular to its stream in 200 seconds. He also finds a man on a raft floating at speed of 3 m/s with river. The distance travelled by boat as seen by man on the raft in crossing the river is -

Options

A. 800 m
B. 1000 m
C. 1200 m
D. 1600 m

Answer

To solve the problem step-by-step, we will analyze the motion of the boat and the raft separately and then combine the information to find the distance traveled by the boat as seen by the man on the raft.

### Step 1: Determine the speed of the boat with respect to the ground.
The width of the river is given as 800 meters, and the time taken to cross it is 200 seconds. We can calculate the speed of the boat using the formula:

\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]

Substituting the values:

\[ \text{Speed of the boat} = \frac{800 \, \text{m}}{200 \, \text{s}} = 4 \, \text{m/s} \]

### Step 2: Determine the speed of the raft.
The speed of the raft is given as 3 m/s. This speed is in the direction of the river's current.

### Step 3: Calculate the velocity of the boat with respect to the raft.
To find the velocity of the boat as seen by the man on the raft, we need to consider the velocities as vectors. The boat is moving perpendicular to the current (in the y-direction), while the raft is moving with the current (in the x-direction).

- Velocity of the boat with respect to the ground: \( \vec{v}_{b} = 4 \, \hat{j} \, \text{m/s} \)
- Velocity of the raft with respect to the ground: \( \vec{v}_{r} = 3 \, \hat{i} \, \text{m/s} \)

The velocity of the boat with respect to the raft is given by:

\[ \vec{v}_{br} = \vec{v}_{b} - \vec{v}_{r} = 4 \, \hat{j} - 3 \, \hat{i} \]

### Step 4: Calculate the magnitude of the velocity of the boat with respect to the raft.
To find the magnitude of the velocity vector \( \vec{v}_{br} \):

\[ |\vec{v}_{br}| = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \, \text{m/s} \]

### Step 5: Calculate the distance traveled by the boat as seen by the man on the raft.
Now, we can find the distance traveled by the boat as observed by the man on the raft. The time taken to cross the river is still 200 seconds.

Using the formula for distance:

\[ \text{Distance} = \text{Speed} \times \text{Time} \]

Substituting the values:

\[ \text{Distance} = 5 \, \text{m/s} \times 200 \, \text{s} = 1000 \, \text{m} \]

### Final Answer:
The distance traveled by the boat as seen by the man on the raft is **1000 meters**.

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