91Ó°ÊÓ

The velocity vector is a crucial concept in kinematics because it tells us how the position of an object changes over time. It encapsulates both speed and direction, making it a vector quantity. In our given exercise, the position function of the dragonfly is expressed as a combination of a quadratic and a cubic term for the respective components in the i and j directions.
To find the velocity vector, we use differentiation with respect to time, which effectively shows us the instantaneous rate of change of the position—essentially how fast and in what direction the dragonfly is moving at any given moment. By differentiating the position vector \( \vec{r} \), we obtain the velocity vector \( \vec{v}(t) \), which is given by:
\[ \vec{v}(t) = (0.180 \text{ m/s}^2 \cdot t) \hat{u} - (0.0450 \text{ m/s}^3 \cdot t^2) \hat{J} \]
This equation indicates that the velocity has two components: one along the \( \hat{u} \) direction that changes linearly with time, and another along the \( \hat{J} \) direction that changes quadratically.

Differentiation is a mathematical technique used to find the rate at which one quantity changes with respect to another. In the realm of kinematics, it is used to move from one type of vector—like position—to another—like velocity or acceleration.
To differentiate the position function, we take the derivative of each term within the vector separately, with respect to time \( t \). The power rule of differentiation is applied here: when differentiating a term \( at^n \), the derivative is \( nat^{n-1} \).
For the given function, \( 2.90 \) is a constant; thus, its derivative is zero. The term \( (0.0900 \text{ m/s}^2) t^2 \) differentiates to \( 2 \times 0.0900 \text{ m/s}^2 \cdot t \), and the term \( -0.0150 \text{ m/s}^3 \cdot t^3 \) differentiates to \( -3 \times 0.0150 \text{ m/s}^3 \cdot t^2 \).
These differentiated terms give us the velocity vector, which provides insight into how the position changes over time.

The acceleration vector represents how the velocity of an object changes over time. In essence, it describes the change in speed or direction of an object. Just as with velocity, acceleration is derived by taking the derivative of the velocity vector with respect to time.
In the context of this exercise, once the velocity vector is determined, differentiating it with respect to time \( t \) gives us the acceleration vector \( \vec{a}(t) \). It is calculated as follows:
For the \( \hat{u} \) component, the derivative of \( 0.180 \text{ m/s}^2 \cdot t \) is \( 0.180 \text{ m/s}^2 \); while for the \( \hat{J} \) component, the derivative of \( -0.0450 \text{ m/s}^3 \cdot t^2 \) is \( -2 \times 0.0450 \text{ m/s}^3 \cdot t = -0.0900 \text{ m/s}^2 \cdot t \).
Thus, the acceleration vector \( \vec{a}(t) \) reflects the linear and quadratic changes in the speed of the dragonfly along the \( \hat{u} \) and \( \hat{J} \) axes.

The position function in kinematics describes where an object is located at a given time. This function is central to understanding motion because it not only reveals the current position but also informs the velocity and acceleration when differentiated appropriately.
For our dragonfly, the position is given as:
\[ \vec{r} = [2.90 \text{ m} + (0.0900 \text{ m/s}^2) t^2] \hat{u} - (0.0150 \text{ m/s}^3) t^3 \hat{J} \]
Each component (\( \hat{u} \) and \( \hat{J} \)) contributes uniquely to the dragonfly's overall position. The quadratic term in the \( \hat{u} \) direction models a motion with increasing velocity, while the cubic term in the \( \hat{J} \) direction suggests an increase in the rate of velocity change over time.

• The quadratic term \( (0.0900 \text{ m/s}^2) t^2 \) indicates motion affected by uniform acceleration.
• The cubic term \( -(0.0150 \text{ m/s}^3) t^3 \) represents a more complex change in motion, possibly due to a varying force or influence.

These aspects of the position function offer insights into both current and future states of motion for any point in time, making it an essential tool in analyzing and predicting movement patterns.