Use the position function s(t)=-16 t^2+v0 t+s0 for...

Use the position function $s(t)=-16 t^{2}+v_{0} t+s_{0}$ for free-falling objects. A silver dollar is dropped from the top of a building that is 1362 feet tall. (a) Determine the position and velocity functions for the coin. (b) Determine the average velocity on the interval [1,2]. (c) Find the instantaneous velocities when $t=1$ and $t=2$. (d) Find the time required for the coin to reach ground level. (e) Find the velocity of the coin at impact.

Use the position function $s(t)=-16 t^{2}+v_{0} t+s_{0}$ for free-falling objects.
A silver dollar is dropped from the top of a building that is
1362 feet tall.
(a) Determine the position and velocity functions for the coin.
(b) Determine the average velocity on the interval [1,2].
(c) Find the instantaneous velocities when $t=1$ and $t=2$.
(d) Find the time required for the coin to reach ground level.
(e) Find the velocity of the coin at impact.

Key Concepts

Instantaneous Velocity

Instantaneous velocity is the velocity of an object at a specific point in time and is found by evaluating the derivative of the position function at that time. It gives a precise measure of the object's speed and direction at that moment.

Solving Quadratic Equations in Kinematics

Determining specific times, such as when an object reaches the ground, often requires solving quadratic equations derived from the position function. This involves finding the roots of the quadratic and interpreting the physically meaningful solution, typically the positive time value.

Average Velocity

Average velocity over a given time interval is calculated as the change in position divided by the change in time. It provides a measure of the overall rate of motion between two time points, not accounting for variations in speed during the interval.

Derivative as a Tool to Determine Velocity

In kinematics, the velocity of an object is obtained by taking the derivative of its position function with respect to time. This mathematical process, rooted in calculus, provides the instantaneous rate of change of position, reflecting the object's speed and direction at any given moment.

Position Function in Free Fall

The position function, typically expressed as s(t) = -16t² + v?t + s? in the imperial system, represents the height of an object as a function of time. It includes terms for gravitational acceleration, initial velocity, and initial height, embodying the parabolic path of a free-falling object.

Free Fall Kinematics

This concept involves the study of motion under the influence of gravity where the only force acting is gravitational force, leading to a uniform acceleration (approximately -32 feet per second squared in imperial units). It forms the basis for analyzing how objects fall and the equations governing their motion.

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