91Ó°ÊÓ

In kinematics, the position vector is essential for determining a particle's location in space over time. The exercise deals with a particle moving in a two-dimensional plane with the given initial position vector: \( \vec{r} = (20.0 \text{ m}) \hat{i} + (40.0 \text{ m}) \hat{j} \). Here, \( \hat{i} \) and \( \hat{j} \) are unit vectors along the horizontal \( x \)-axis and vertical \( y \)-axis, respectively. To find the position vector at any time \( t \), we first obtain the velocity function by integrating the acceleration function. Once the velocity is known, integrating it will give the position function. The constants of integration, \( D_1 \) and \( D_2 \), ensure the position function aligns with the initial position at \( t=0 \). This confirms that the process accurately reflects real motion starting from position \( 20 \hat{i} + 40 \hat{j} \) at \( t=0 \). The importance of filing these integrations correctly is to predict the path of the particle accurately. Using the derived position function, we can substitute \( t=4 \) s to find the particle's position at that moment as \( 72 \hat{i} + 90.67 \hat{j} \). This illustrates not only where the particle started but how it traveled over time.

The velocity function describes how fast and in which direction a particle is moving at any given time. Velocity is derived from the acceleration by integration, as seen when we find \( \vec{v}(t) = \int \vec{a} \, dt \).Here, the acceleration \( \vec{a} = 3t \hat{i} + 4t \hat{j} \) shows that both \( x \) and \( y \) components depend linearly on time \( t \).**Steps to Obtain the Velocity Function**:

• Integrate each component of the acceleration separately.
• For \( \,i\,-component: \int 3t \, dt = \frac{3}{2}t^2 + C_1 \).
• For \( \,j\,-component: \int 4t \, dt = 2t^2 + C_2 \).

The constants \( C_1 \) and \( C_2 \), determined using initial conditions \( \vec{v}(0) = 5 \hat{i} + 2 \hat{j} \), give us the complete velocity function: \( \vec{v}(t) = \left( \frac{3}{2}t^2 + 5 \right) \hat{i} + \left( 2t^2 + 2 \right) \hat{j} \).Understanding the velocity function is critical, as it informs us of the changes in the particle's speed and direction over time. At \( t=4 \) s, the velocity function becomes \( 29 \hat{i} + 34 \hat{j} \), essential for analyzing the particle's motion and making predictions.

Acceleration is a fundamental concept in kinematics that informs us how the velocity of a particle changes over time. In the given exercise, the acceleration vector \( \vec{a} = 3t \hat{i} + 4t \hat{j} \) represents a uniformly changing motion, meaning both its \( x \)-component and \( y \)-component magnitude increase linearly with time.The components are:

• \( 3t \hat{i} \): indicating the acceleration in the horizontal direction.
• \( 4t \hat{j} \): indicating the acceleration in the vertical direction.

Notice how the linear dependency on time \( t \) tells us the rate at which the velocity is changing, essentially describing a parabolic trajectory in the \( xy \)-plane.By integrating these linear components, we form the velocity function, and a second integration gives us the position function. The acceleration forms the basis from which we derive all other motion-related functions in this exercise.Understanding acceleration as the rate of change of velocity helps identify how quickly a particle picks up speed or slows down, playing a pivotal role in predicting future motion paths.