In a river , a boat with a vertical pole on it is moving and velocity of river, boat and man climbing on the pole are given below
`barV_(1)=3 m//s hati,barV_j= 4m//s hatj,V_(mb)= 2m//s k`
Symbols r,b and m are used for river boat and man.Velocity of the man as observed by ground observer will be

Options

A. `3hati+4hatj+2hatk`
B. `4hatj+2hatk`
C. `2hatk`
D. `3hati+2hatk`

Explanation

To find the velocity of the man climbing on the pole as observed by a ground observer, we need to consider the velocities of the river, the boat, and the man in their respective directions.

### Step-by-Step Solution:

1. **Identify the given velocities:**
- Velocity of the river, \( \bar{V}_r = 3 \, \text{m/s} \, \hat{i} \) (in the x-direction)
- Velocity of the boat, \( \bar{V}_b = 4 \, \text{m/s} \, \hat{j} \) (in the y-direction)
- Velocity of the man with respect to the boat, \( \bar{V}_{mb} = 2 \, \text{m/s} \, \hat{k} \) (in the z-direction)

2. **Understand the reference frames:**
- The velocity of the boat is given in the y-direction, and the man is climbing vertically in the z-direction. The river's velocity is in the x-direction.

3. **Calculate the velocity of the man as observed from the ground:**
- The velocity of the man with respect to the ground can be found by adding the velocity of the man with respect to the boat and the velocity of the boat with respect to the ground.
- Mathematically, this can be expressed as:
\[ \bar{V}_m = \bar{V}_{mb} + \bar{V}_b \]
- Substitute the values:
\[ \bar{V}_m = (2 \, \hat{k}) + (4 \, \hat{j}) \]

4. **Combine the vectors:**
- Since the velocities are in different directions, we can write:
\[ \bar{V}_m = 4 \, \hat{j} + 2 \, \hat{k} \]

5. **Final result:**
- The velocity of the man as observed by a ground observer is:
\[ \bar{V}_m = 4 \, \hat{j} + 2 \, \hat{k} \, \text{m/s} \]