91Ó°ÊÓ

Understanding displacement is pivotal when learning about motion in physics. The displacement of an object is a vector quantity that refers to the change in position of the object. It is important to distinguish displacement from distance. Displacement takes into account the direction of movement and is the straight line path between the initial and final positions.

To calculate displacement, we use the simple formula \( \Delta \vec{r} = \vec{r}_f - \vec{r}_i \), where \(\vec{r}_i\) and \(\vec{r}_f\) represent the initial and final position vectors of the squirrel, respectively. In our case, the squirrel has the initial coordinates \( (1.1m, 3.4m) \) and final coordinates \( (5.3m, -0.5m) \), hence \( \Delta \vec{r} \) becomes \( (5.3m - 1.1m, -0.5m - 3.4m) \).

The displacement in two dimensions involves subtraction of the individual components, so we end up with \( \Delta x \) and \( \Delta y \), representing the displacement along the x-axis and y-axis, respectively. Calculating the displacement correctly is crucial for subsequent steps, such as finding the velocity components or vector magnitude.

Velocity, a vector quantity, has two key components, one for each direction in a two-dimensional plane: \( v_x \) for the horizontal (x) component and \( v_y \) for the vertical (y) component. We calculate these by dividing the displacement by the total time taken for that displacement. This calculation represents the average velocity components over the time interval considered.

To find the average velocity components for our furry friend, the formula \( v_x = \frac{\Delta x}{\Delta t} \) is used for the x-direction, and \( v_y = \frac{\Delta y}{\Delta t} \) is applied for the y-direction, where \( \Delta t \) is the time difference \( t_2 - t_1 \). Plugging in the numbers, we compute the average velocities for both the horizontal and vertical motions separately. These components are crucial as they lead us to understanding the overall movement of the squirrel through the magnitude and the direction of its average velocity.

Once we have the velocity components, we can proceed to find the vector magnitude and direction of the squirrel’s average velocity. The magnitude is essentially the length of the velocity vector and represents how fast the squirrel is moving irrespective of the direction. It is calculated using the Pythagorean theorem due to the perpendicular nature of the velocity components. The formula for magnitude is \( v = \sqrt{v_{x}^2 + v_{y}^2} \). This value gives a clear and singular representation of the velocity without complicating the direction.

The direction of the average velocity vector informs us about which way the squirrel is moving. To find the direction, we use the arctangent function, denoted as arctan or tan^-1, and apply it to the ratio of the y-component to the x-component of the velocity: \( \theta = arctan(\frac{v_{y}}{v_{x}}) \). It is important to consider the signs of the velocity components when determining the direction as this influences the quadrant in which the vector lies. For our case, since the y-component is negative, the angle \( \theta \) will lie in the fourth quadrant, indicative of south-east direction relative to the positive x-axis.