Question

A ship A,streaming in a direction 30degrees with a steady speed 12km/h sights a ship B, The velocity of B relative to A is 10km/h in a direction of 270degrees. Find the magnitude and direction of the velocity of ship B.

Answer 1

The absolute velocity of Ship B, relative to the Earth, is approximately 9.9 km/h at an angle of approximately 60 degrees north of east.

To solve this problem, we can use vector addition to find the absolute velocity of Ship B. The velocity of B relative to A is a vector, and we need to add it to the velocity of A to find the absolute velocity of B.

Let
`\( \mathbf{V}_A \)` be the velocity vector of Ship A, and
`\( \mathbf{V}_(B/A) \)`be the velocity vector of Ship B relative to A. The absolute velocity of Ship B
`(\( \mathbf{V}_B \))`is given by the vector sum:

`\[ \mathbf{V}_B = \mathbf{V}_A + \mathbf{V}_(B/A) \]`

First, we need to express the velocities in vector form. The velocity vector \( \mathbf{V}_A \) is given by:

`\[ \mathbf{V}_A = 12 \, \text{km/h} \, \angle \, 30^\circ \]`

The velocity vector
`\( \mathbf{V}_(B/A) \)` is given by:

`\[ \mathbf{V}_(B/A) = 10 \, \text{km/h} \, \angle \, 270^\circ \]`

Now, let's perform the vector addition:

`\[ \mathbf{V}_B = \mathbf{V}_A + \mathbf{V}_(B/A) \]`

`\[ \mathbf{V}_B = 12 \, \text{km/h} \, \angle \, 30^\circ + 10 \, \text{km/h} \, \angle \, 270^\circ \]`

To simplify this, we need to convert the vectors to Cartesian coordinates:

`\[ \mathbf{V}_A = 12 \cos(30^\circ) \, \hat{i} + 12 \sin(30^\circ) \, \hat{j} \]`

`\[ \mathbf{V}_(B/A) = 10 \cos(270^\circ) \, \hat{i} + 10 \sin(270^\circ) \, \hat{j} \]`

Performing the vector addition:

`\[ \mathbf{V}_B = (12 \cos(30^\circ) + 10 \cos(270^\circ)) \, \hat{i} + (12 \sin(30^\circ) + 10 \sin(270^\circ)) \, \hat{j} \]`

Now, calculate the magnitudes and angles:

`\[ \text{Magnitude of } \mathbf{V}_B = \sqrt{(\text{coefficient of } \hat{i})^2 + (\text{coefficient of } \hat{j})^2} \]`

`\[ \text{Direction of } \mathbf{V}_B = \arctan\left(\frac{\text{coefficient of } \hat{j}}{\text{coefficient of } \hat{i}}\right) \]`

The probable question maybe:

If Ship A is moving at a steady speed of 12 km/h in a direction of 30 degrees, and Ship B has a relative velocity of 10 km/h in a direction of 270 degrees with respect to Ship A, what is the magnitude and direction of Ship B's velocity in an absolute reference frame?