91Ó°ÊÓ

Vector acceleration is an important concept in physics which describes how the velocity of an object changes with time, in both magnitude and direction. It is crucial for understanding motion as it underpins the dynamics of objects. In the given problem, the dog's acceleration is expressed as a vector \( \vec{a}(t) \). The vector has both an x-component in the direction \( \hat{i} \) and a y-component in the direction \( \hat{j} \.\)

The function \( \vec{a}(t) = (0.400 \,m/s^2) \hat{i} - (0.180 \,m/s^3) t \hat{j} \) signifies that the dog's acceleration in the x-direction is constant, while in the y-direction it decreases linearly over time. This variable acceleration means that the dog’s velocity will not simply increase at a steady rate; instead, its behavior over time will be more complex due to the changing y-component.

Velocity integration is the process by which we determine an object’s velocity based on its acceleration. Since acceleration is the rate of change of velocity, by integrating the acceleration over time, we determine the velocity function.

For the problem at hand, the acceleration vector will be integrated component-wise to find the velocity. The integration of the constant x-acceleration yields a linear velocity function \(v_i(t) = (0.400 \, m/s^2) t\). On the other hand, integrating the time-dependent y-acceleration gives a quadratic term \(v_j(t) = - (0.180 \, m/s^3) t^2 / 2\). Thus, integrating the acceleration vector is not merely about calculating an area under a curve; it establishes the foundation of how the dog's velocity changes over time.

The position function describes the location of an object as a function of time. After determining velocity through integration of acceleration, we integrate the velocity to get the position function. The position function is essential for interpreting an object's movement over time and determining its location at any point.

For the running dog, further integration of the velocity components obtained from the previous step provides the x- and y-position functions \(r_i(t) = (0.400 \, m/s^2) t^2 / 2\) and \(r_j(t) = - (0.180 \, m/s^3) t^3 / 6\), respectively. These equations describe the dog's path in two dimensions and will allow us to calculate how far the dog has traveled from its starting point.

The magnitude of displacement is a scalar value representing the distance from an object's starting point to its final position, regardless of its path traveled. It is found by determining the length of the displacement vector, which consists of the combined x- and y-components of the dog's position.

In our scenario, the dog starts at the origin, and after 8 seconds, it has position components \(r_i(8.00 s)\) and \(r_j(8.00 s)\). The magnitude of displacement is calculated using the Pythagorean theorem. For the dog, the formula yields \(d(8.00 s) = \sqrt{r_i(8.00 s)^2 + r_j(8.00 s)^2}\). This calculation elucidates the direct distance the dog is from the tree after the elapsed time, providing a clear measure of its overall displacement.