91Ó°ÊÓ

Vector addition is a crucial concept when dealing with relative velocities. It allows us to determine the net velocity of an object as seen by another observer. In the exercise, Barbara and Neil's motions can be represented as vectors. Barbara, moving south, has a velocity vector of

• \( \mathbf{v_B} = (0, -4.0) \text{ m/s} \)

Please note that the south direction is represented as the negative y-direction. Meanwhile, Neil moves west, which results in the following vector:

• \( \mathbf{v_N} = (-3.2, 0) \text{ m/s} \)

Thus, west is represented as the negative x-direction.

To find their relative velocity, these vectors are subtracted. The formula

• \( \mathbf{v_{N/B}} = \mathbf{v_N} - \mathbf{v_B} \)

is applied, resulting in the relative velocity vector

• \( \mathbf{v_{N/B}} = (-3.2, 4.0) \text{ m/s} \)

By understanding how to properly set up and manipulate these vectors, you're able to comprehend how one object's motion appears from another object's frame of reference.

The Pythagorean theorem is fundamental in finding the magnitude of a vector. When you have a vector expressed by its components, the theorem helps compute its length. For the relative velocity vector

• \( \mathbf{v_{N/B}} = (-3.2, 4.0) \text{ m/s} \)

the magnitude is calculated by treating these components as sides of a right triangle. This is because:

• The x-component \((-3.2)\)
• The y-component \((4.0)\)

represent the leg lengths.

The formula is:

• \[ \| \mathbf{v_{N/B}} \| = \sqrt{(-3.2)^2 + 4.0^2} \]

This results in the magnitude:

• \( \approx 5.12 \text{ m/s} \)

Through the Pythagorean theorem, you can determine the true speed of Neil's velocity relative to Barbara. Each component measures the effect of velocity in either direction, and when combined, they form a single magnitude.

To understand the direction of a vector, we need to calculate the angle it forms with a reference direction. For the velocity vector

• \( \mathbf{v_{N/B}} = (-3.2, 4.0) \text{ m/s} \)

the reference direction is west, or along the x-axis. We find the angle between the vector and the west direction using trigonometric functions. This helps in expressing Neil's velocity relative to Barbara as being not just a magnitude but also a clear direction.

To calculate this angle \( \theta \), we use:

• \( \theta = \tan^{-1} \left( \frac{4.0}{3.2} \right) \)

This leads to:

• \( \theta \approx 51.3^\circ \)

Thus, the direction is 51.3 degrees north of west. Angle calculation integrates information about vector direction, offering a complete understanding beyond magnitude.

Understanding the tangent function is fundamental in angle calculation. It comes from trigonometry and is used in problems involving right triangles. For our exercise, it helps determine the angle of the relative velocity vector. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Here:

• The opposite side \( = 4.0 \text{ m/s} \)
• The adjacent side \( = 3.2 \text{ m/s} \)

tangent then becomes:

• \( \tan(\theta) = \frac{4.0}{3.2} \)

To find \( \theta \), we compute the inverse tangent (or arctan):

• \( \theta = \tan^{-1} \left( \frac{4.0}{3.2} \right) \)

This calculation demonstrates how the tangent function is a tool that converts a relationship between sides of triangles into a practical measure of direction between vectors.