Vertical Motion The height of an object moving vertically is...

Vertical Motion The height of an object moving vertically is
given by $$s=-16 t^{2}+96 t+112$$
with $s$ in $\mathrm{ft}$ and $t$ in sec. Find (a) the object's velocity when
$t=0,$ (b) its maximum height and when it occurs, and
(c) its velocity when $s=0$ .

Key Concepts

Kinematics in Vertical Motion

This concept pertains to the study of motion in a straight line under the influence of gravity, where the position of an object is described by a function of time. The equation for vertical motion typically includes terms for gravitational acceleration, initial velocity, and initial displacement, allowing the analysis of key motion characteristics such as time of ascent, maximum height, and impact time.

Differentiation to Determine Velocity

In kinematics, the velocity of an object is found by differentiating its displacement function with respect to time. This process provides the instantaneous rate of change of position, which is crucial for understanding how quickly the object is moving at any given moment.

Vertex of a Quadratic Function

A quadratic function representing vertical motion opens downward when the coefficient of the squared term is negative. The vertex of this parabola indicates the maximum point, which corresponds to the highest height reached by the object, as well as the exact time when this occurs.

Solving Quadratic Equations

To find specific times during the motion, such as when the object reaches the ground, the displacement function is set to zero and solved as a quadratic equation. This process is essential for determining the roots, one of which will indicate the relevant physical time for the event.

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