Vertical motion The height above ground of an object moving...

Vertical motion The height above ground of an object moving vertically is given by \begin{equation}s=-16 t^{2}+96 t+112,\end{equation} with $s$ in feet and $t$ in seconds. Find \begin{equation} \begin{array}{l}{\text { a. the object's velocity when } t=0;} \\ {\text { b. its maximum height and when it occurs; }} \\ {\text { c. its velocity when } s=0 \text {.}}\end{array} \end{equation}

Vertical motion The height above ground of an object moving vertically is given by
\begin{equation}s=-16 t^{2}+96 t+112,\end{equation}
with $s$ in feet and $t$ in seconds. Find
\begin{equation}
\begin{array}{l}{\text { a. the object's velocity when } t=0;} \\ {\text { b. its maximum height and when it occurs; }} \\ {\text { c. its velocity when } s=0 \text {.}}\end{array}
\end{equation}

Key Concepts

Vertical Motion under Gravity

This concept involves analyzing the movement of objects influenced by the constant acceleration due to gravity. In such scenarios, the motion is typically modeled by quadratic equations because the position of the object as a function of time is a parabolic curve, reflecting the uniform acceleration acting downward.

Derivatives in Kinematics

This key concept deals with using calculus to determine how the position of an object changes over time. The derivative of the position function with respect to time provides the velocity, offering insight into the instantaneous rate of change and allowing analysis of the object's speed and direction at any given moment.

Quadratic Functions and Their Properties

Quadratic functions are used to represent the path of projectiles and other objects under uniform acceleration. Essential properties include the vertex, which represents the maximum or minimum value of the function, and the symmetry of the parabola, which aids in understanding the timing and nature of specific events during the motion, such as reaching the peak height.

Solving Quadratic Equations

This concept is crucial when determining the specific times at which an event occurs, such as when an object reaches a certain height. Techniques for solving quadratic equations—like using the quadratic formula, factoring, or completing the square—enable one to find the roots of the equation, which correspond to key moments in the object’s trajectory.

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