A ship is steaming towards east with a speed of 8 m/s. A women runs across the deck at a speed of `6 ms^(-1)` towards north. What is the velocity of the women relative to the sea ?

Answer

To find the velocity of the woman relative to the sea, we can use vector addition. Here’s a step-by-step solution:

### Step 1: Identify the velocities
- The velocity of the ship (V_s) is 8 m/s towards the east.
- The velocity of the woman (V_w) is 6 m/s towards the north.

### Step 2: Represent the velocities as vectors
- We can represent the velocity of the ship as a vector:
\( \vec{V_s} = 8 \hat{i} \) m/s (where \( \hat{i} \) is the unit vector in the east direction).
- The velocity of the woman can be represented as:
\( \vec{V_w} = 6 \hat{j} \) m/s (where \( \hat{j} \) is the unit vector in the north direction).

### Step 3: Find the resultant velocity of the woman relative to the sea
The velocity of the woman relative to the sea (V_w relative to sea) can be calculated using vector addition:
\[ \vec{V_{w, sea}} = \vec{V_w} + \vec{V_s} \]
Substituting the vectors:
\[ \vec{V_{w, sea}} = 6 \hat{j} + 8 \hat{i} \]

### Step 4: Calculate the magnitude of the resultant velocity
To find the magnitude of the resultant velocity, we use the Pythagorean theorem:
\[ |\vec{V_{w, sea}}| = \sqrt{(V_s)^2 + (V_w)^2} \]
Substituting the values:
\[ |\vec{V_{w, sea}}| = \sqrt{(8)^2 + (6)^2} \]
\[ |\vec{V_{w, sea}}| = \sqrt{64 + 36} \]
\[ |\vec{V_{w, sea}}| = \sqrt{100} \]
\[ |\vec{V_{w, sea}}| = 10 \text{ m/s} \]

### Step 5: Determine the direction of the resultant velocity
To find the direction, we can calculate the angle θ with respect to the east (x-axis) using the tangent function:
\[ \tan(\theta) = \frac{V_w}{V_s} \]
\[ \tan(\theta) = \frac{6}{8} \]
\[ \theta = \tan^{-1}\left(\frac{6}{8}\right) \]
\[ \theta = \tan^{-1}(0.75) \]
Calculating this gives approximately:
\[ \theta \approx 36.87^\circ \]

### Final Answer
The velocity of the woman relative to the sea is approximately **10 m/s at an angle of 36.87° north of east**.

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