A ship is sailing due north at a speed of `1.25 ms^-1`. The current is taking it towards east at the rate of `2 ms^-1`. and a sailor is climbing a vertical pole in the ship at the rate of `0.25 ms^-1`. Find the magnitude of the velocity of the sailor with respect to ground.

Explanation

To find the magnitude of the velocity of the sailor with respect to the ground, we need to consider the velocities of the ship, the current, and the sailor climbing the pole. Let's break it down step by step.

### Step 1: Identify the velocities
1. The ship is sailing due north at a speed of \(1.25 \, \text{ms}^{-1}\). This can be represented as a vector:
\[ \vec{V}_{\text{ship}} = 0 \hat{i} + 1.25 \hat{j} + 0 \hat{k} \]
(where \(\hat{i}\) is east, \(\hat{j}\) is north, and \(\hat{k}\) is vertical).

2. The current is taking the ship towards the east at a rate of \(2 \, \text{ms}^{-1}\):
\[ \vec{V}_{\text{current}} = 2 \hat{i} + 0 \hat{j} + 0 \hat{k} \]

3. The sailor is climbing the vertical pole at a rate of \(0.25 \, \text{ms}^{-1}\):
\[ \vec{V}_{\text{sailor}} = 0 \hat{i} + 0 \hat{j} + 0.25 \hat{k} \]

### Step 2: Combine the velocities
To find the total velocity of the sailor with respect to the ground, we need to add the vectors:
\[ \vec{V}_{\text{sailor, ground}} = \vec{V}_{\text{ship}} + \vec{V}_{\text{current}} + \vec{V}_{\text{sailor}} \]
Substituting the vectors we have:
\[ \vec{V}_{\text{sailor, ground}} = (0 + 2 + 0) \hat{i} + (1.25 + 0 + 0) \hat{j} + (0 + 0 + 0.25) \hat{k} \]
This simplifies to:
\[ \vec{V}_{\text{sailor, ground}} = 2 \hat{i} + 1.25 \hat{j} + 0.25 \hat{k} \]

### Step 3: Calculate the magnitude of the velocity
The magnitude of the velocity vector can be calculated using the formula:
\[ |\vec{V}| = \sqrt{(V_x)^2 + (V_y)^2 + (V_z)^2} \]
Substituting the components of the velocity vector:
\[ |\vec{V}_{\text{sailor, ground}}| = \sqrt{(2)^2 + (1.25)^2 + (0.25)^2} \]
Calculating each term:
\[ = \sqrt{4 + 1.5625 + 0.0625} \]
\[ = \sqrt{5.625} \]
Calculating the square root:
\[ = \sqrt{5.625} \approx 2.37 \, \text{ms}^{-1} \]

### Final Answer
The magnitude of the velocity of the sailor with respect to the ground is approximately \(2.37 \, \text{ms}^{-1}\).

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