91Ó°ÊÓ

Relative velocity is the measure of the velocity of one object as observed from another moving object. In our problem, we calculate the velocity of the soccer ball relative to Mia, who is also in motion. This approach is useful in scenarios where both the observer and the object of interest are moving.

In the context of this exercise, you first determine the velocity of the ball with respect to Mia. Then, you use this information to find the velocity of the ball relative to the ground. The relative velocity vector, in this case, becomes a crucial step in determining the final answer. It takes into account both Mia's movement and the ball's path relative to her, allowing us to calculate the absolute velocity of the ball in relation to a fixed point, which is the ground in this scenario.

Breaking down a velocity into its components is essential for analyzing its effect in different directions. In vector analysis, we understand movements in terms of north-south (vertical or y-direction) and east-west (horizontal or x-direction).

For this problem, the ball's velocity relative to Mia was given at an angle. To simplify calculations, we split this velocity into two perpendicular components:

• Southward (vertical): calculated using cosine for the southern component.
• Eastward (horizontal): calculated using sine for the eastern component.

By separating these components, we enable easier integration of this velocity with Mia’s velocity to find the total velocity from the perspective of the ground.

The Pythagorean theorem is a fundamental principle used to calculate the magnitude of a resultant vector when its components are known. In the context of our exercise, we were given the components of the ball's velocity with respect to the ground.

The formula used is: \[\| \vec{v}_B \| = \sqrt{(2.50)^2 + (1.67)^2}\]This formula helps us find the absolute magnitude of the resultant velocity vector by applying the square root of the sums of the squared components. This step is crucial to determine how fast the ball is moving from the ground's point of view. This step gives the result in meters per second, providing a clear understanding of the ball's speed.

The tangent ratio is a trigonometric function used to find angles when two sides of a right triangle are known. It is particularly useful in vector analysis when determining the direction of a vector.

For this exercise, once the components of the velocity vector were known, you can determine the direction angle \(\theta\) using:\[\tan \theta = \frac{2.50}{1.67}\]The angle \(\theta\) is then found using the inverse tangent function:\[\theta = \tan^{-1} \left(\frac{2.50}{1.67}\right)\]This process results in an angle within the first quadrant, demonstrating how the tangent ratio helps in defining direction in polar form from known vector components. This allows you to present the velocity not just as a magnitude but also as a direction relative to a given axis, enhancing the understanding of the ball’s trajectory.