An aeroplane has a velocity of 110 m/s directed due north and is subjected to a wind blowing from west to east at a speed of 40 m/s . Calculate the actual velocity of the aeroplane relative to the earth .

Explanation

To solve the problem of finding the actual velocity of the aeroplane relative to the earth, we can follow these steps:

### Step 1: Identify the velocities
- The velocity of the aeroplane relative to the wind is given as \( V_{a/w} = 110 \, \text{m/s} \) directed due north.
- The velocity of the wind is given as \( V_{w} = 40 \, \text{m/s} \) directed from west to east.

### Step 2: Represent the velocities as vectors
- We can represent the velocity of the aeroplane as a vector pointing north:
\[ \vec{V}_{a/w} = (0, 110) \, \text{m/s} \]
where the first component is the east-west direction (x-axis) and the second component is the north-south direction (y-axis).

- The velocity of the wind can be represented as a vector pointing east:
\[ \vec{V}_{w} = (40, 0) \, \text{m/s} \]

### Step 3: Calculate the resultant velocity
- The actual velocity of the aeroplane relative to the earth \( \vec{V}_{a/g} \) can be found by adding the two vectors:
\[ \vec{V}_{a/g} = \vec{V}_{a/w} + \vec{V}_{w} = (0, 110) + (40, 0) = (40, 110) \, \text{m/s} \]

### Step 4: Calculate the magnitude of the resultant velocity
- The magnitude of the resultant velocity can be calculated using the Pythagorean theorem:
\[ |\vec{V}_{a/g}| = \sqrt{(40)^2 + (110)^2} \]
\[ |\vec{V}_{a/g}| = \sqrt{1600 + 12100} = \sqrt{13700} \approx 117.04 \, \text{m/s} \]

### Step 5: Calculate the direction of the resultant velocity
- To find the angle \( \theta \) that the resultant velocity makes with the north direction, we can use the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{40}{110} \]
- Therefore,
\[ \theta = \tan^{-1}\left(\frac{40}{110}\right) \approx 20^\circ \]

### Final Result
- The actual velocity of the aeroplane relative to the earth is approximately \( 117.04 \, \text{m/s} \) at an angle of \( 20^\circ \) east of north.

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