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To solve the problem shown in the image, we need to determine the velocity of object \( A \) with respect to object \( B \).

Given:
- Velocity of \( A \) (denoted as \( \vec{v}_A \)) = \( 5 \, \text{m/s} \) (along the horizontal axis)
- Velocity of \( B \) (denoted as \( \vec{v}_B \)) = \( 5 \, \text{m/s} \) at an angle of \( 60^\circ \) with respect to the horizontal axis

### Step 1: Break Down the Velocities into Components
First, break down the velocity of \( B \) into its horizontal and vertical components:
- \( v_{Bx} = v_B \cos(60^\circ) = 5 \times \frac{1}{2} = 2.5 \, \text{m/s} \) (horizontal component)
- \( v_{By} = v_B \sin(60^\circ) = 5 \times \frac{\sqrt{3}}{2} = \frac{5\sqrt{3}}{2} \approx 4.33 \, \text{m/s} \) (vertical component)

### Step 2: Calculate the Relative Velocity
The velocity of \( A \) with respect to \( B \) is given by:
\[
\vec{v}_{AB} = \vec{v}_A - \vec{v}_B
\]

So:
\[
v_{ABx} = v_A - v_{Bx} = 5 \, \text{m/s} - 2.5 \, \text{m/s} = 2.5 \, \text{m/s}
\]
\[
v_{ABy} = 0 - v_{By} = - \frac{5\sqrt{3}}{2} \approx -4.33 \, \text{m/s}
\]

### Step 3: Determine the Magnitude of the Relative Velocity
The magnitude of the relative velocity \( v_{AB} \) is given by:
\[
v_{AB} = \sqrt{(v_{ABx})^2 + (v_{ABy})^2}
\]
\[
v_{AB} = \sqrt{(2.5)^2 + \left(-\frac{5\sqrt{3}}{2}\right)^2}
\]
\[
v_{AB} = \sqrt{6.25 + \left(\frac{25 \times 3}{4}\right)} = \sqrt{6.25 + 18.75} = \sqrt{25} = 5 \, \text{m/s}
\]

Thus, the magnitude of the velocity of \( A \) with respect to \( B \) is \( 5 \, \text{m/s} \).

The correct answer is **(2) \( 5 \, \text{m/s} \)**.

Would you like more details on any part of this problem? Here are some related questions:

1. How do you determine the direction of the relative velocity?
2. What is the significance of the angle in determining velocity components?
3. Can this approach be generalized to 3D motion?
4. How would the result change if the angle was different?
5. What is the physical meaning of relative velocity in this context?

**Tip:** Always break vectors into their components when dealing with problems involving angles; it simplifies the calculations and avoids errors.

In the figure shown, velocity of A with respect to B will be?

Solve the problem of finding the relative velocity of object A with respect to object B using vector decomposition and the Pythagorean theorem. Learn how to break down velocities into components and calculate the magnitude of the relative velocity. Suitable for high school physics students.

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