91Ó°ÊÓ

When dealing with problems involving motion, it's essential to break down any movement into its fundamental parts, or components. In two dimensions, these components are typically along the x-axis (horizontal) and y-axis (vertical). This breakdown helps us better analyze how an object, like a squirrel in our exercise, moves in each direction independently.
To find the average velocity components, you calculate both the individual changes in the x and y directions over a specified time period:

• \( v_{x, \text{avg}} = \frac{\Delta x}{\Delta t} \)
• \( v_{y, \text{avg}} = \frac{\Delta y}{\Delta t} \)

By computing these elements, we determine how fast the squirrel is moving horizontally and vertically between the starting and ending coordinates. This calculates the rate of change of position, or velocity, in their respective directions.

Coordinate geometry allows us to precisely describe the position and trajectory of objects in space. By using a simple (x, y) coordinate system, we can track movement and calculate distances and direction.
For our exercise, we first note the starting point \((1.1 \text{ m}, 3.4 \text{ m})\) and ending point \((5.3 \text{ m}, -0.5 \text{ m})\) of the squirrel's path. Within this framework, the change in position or displacement is readily seen:

• The change in the x-coordinate (horizontal distance) is \(\Delta x\).
• The change in the y-coordinate (vertical distance) is \(\Delta y\).

By calculating these displacements, we easily derive path-related details, such as the velocity components, using these coordinates as a basis. Finally, coordinate geometry emphasizes the relationship between the numerical values and the graphical position, enhancing our understanding of the motion.

Velocity isn't just about how fast something moves; it defines both speed and direction. To fully understand the squirrel's velocity, we must determine its magnitude and direction.Magnitude of Velocity
The magnitude of velocity gives us the overall speed irrespective of direction. Using the velocity components, the magnitude can be calculated through the Pythagorean theorem:

• \[ v_{\text{avg}} = \sqrt{(v_{x, \text{avg}})^2 + (v_{y, \text{avg}})^2} \]

In this example, the computation results in approximately 1.91 m/s, showing the pace at which the squirrel moves along its trajectory.Direction of Velocity
The direction angle \(\theta\) indicates which way the object is heading relative to the positive x-axis. Calculated through the inverse tangent function, the direction provides context to the speed:

• \( \theta = \tan^{-1} \left(\frac{v_{y, \text{avg}}}{v_{x, \text{avg}}}\right) \)

This exercise yields \(-43.6^\circ\), which means the direction is diagonally southward, or below the positive x-axis. Together, the magnitude and direction fully describe the velocity vector, painting a clear picture of the squirrel's movement.