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Particle motion within the context of multi-variable calculus involves understanding how an object, such as a particle, moves within a three-dimensional space over time. This problem deals with a particle whose position is expressed as a vector function of time. The position function gives coordinates for the particle at any time, thus offering a path known as a parametric curve. For example, the vector function \( \vec{r}(t) = 2t \vec{i} + 3t \vec{j} + (100 - (t-5)^2) \vec{k} \) tells us that the particle's position changes over time represented by \( t \).
This equation means the particle's horizontal components are linear with time, moving in vectors \( \vec{i} \) and \( \vec{j} \). However, its vertical motion given by the \( \vec{k} \) component follows a parabolic path. This parabola peaks at time \( t = 5 \), when the derivative of height with respect to time reaches zero. At the peak, the particle achieves its maximum height of 100 cm.
Tracking particle motion through vector functions helps solve intricate challenges in physics and engineering, as it shows how objects traverse space in real time.

Vector-valued functions play a crucial role in describing trajectories, like the particle's path in this problem. Unlike ordinary functions, vector-valued functions have multiple components that show movement in different spatial dimensions.
In this exercise, the vector function \( \vec{r}(t) \) encapsulates motion through \( \vec{i} \), \( \vec{j} \), and \( \vec{k} \). Each part:

• \( 2t \vec{i} \) conveys linear motion along the \( x \)-axis.
• \( 3t \vec{j} \) captures the linear movement along the \( y \)-axis.
• \( (100 - (t-5)^2) \vec{k} \) indicates parabolic motion along the \( z \)-axis.

Vector-valued functions allow us to take derivatives to find velocities, i.e., the particle's speed, by deriving each component with respect to time. As calculated, \( \vec{v}(t) = 2 \vec{i} + 3 \vec{j} + 2(t-5) \vec{k} \). The speed, or the magnitude of this velocity vector, is the square root value derived from the sum of the squares of these components. Understanding vector-valued functions enhances our ability to decode complex trajectories and compute rates of changes parallelly in multiple directions.

Optimization is about finding the best solution within a set of constraints. In this context, it reflects maximizing or minimizing a certain quantity: the particle's height or speed. We engage in optimization to determine the specific time \( t \) that gives either the maximum height of the particle or the speed it achieves at various times.
For the height, we calculated the derivative of the \( z \)-component and set it to zero to find when the height was highest. The second derivative confirmed a maximum at \( t = 5 \). The speed issue follows a similar method: we look for extrema in the speed function, which also led to the conclusion that the speed is maximized at \( t = 5 \). These calculations involve setting derivatives to zero and considering second derivatives to confirm maxima or minima.
Optimization through calculus is valuable as it brings precision to predicting phenomena, crucial in fields from science to economics. It allows one to solve real-world problems by pinpointing times of fastest or slowest speeds or locating the highest achievable points on a path.