If you are on the moon, then an orange falling from 20 feet...

If you are on the moon, then an orange falling from 20 feet has a height of $s(t)=20-2.65 t^{2}$ feet when it has fallen for $t$ seconds. (a) Graph the position function $s(t)$ and find the time that the orange will hit the surface of the moon. (b) Make a table to record the average rates that the orange is falling during the last second, half-second, quarter-second, and eighth-of-a-second of its fall on the moon. (c) From the data in your table, make a guess for the instantaneous final velocity of the orange at the moment it hits the surface of the moon.

If you are on the moon, then an orange falling from 20 feet has a height of $s(t)=20-2.65 t^{2}$ feet when it has fallen for $t$ seconds.
(a) Graph the position function $s(t)$ and find the time that the orange will hit the surface of the moon.
(b) Make a table to record the average rates that the orange is falling during the last second, half-second, quarter-second, and eighth-of-a-second of its fall on the moon.
(c) From the data in your table, make a guess for the instantaneous final velocity of the orange at the moment it hits the surface of the moon.

Key Concepts

Average Rate of Change

The average rate of change quantifies how a function's output changes over a specific interval of time. In kinematics, it is used to approximate the object's velocity over discrete intervals, providing insight into its overall behavior before landing.

Graphing Functions

Graphing functions is a method used to visually represent the behavior of a physical system over time. In the context of motion, plotting the position function helps in understanding how the object's displacement evolves and in identifying critical moments such as impacts or maximum height.

Instantaneous Rate of Change (Derivative)

The instantaneous rate of change, often calculated as the derivative of the position function, represents the object's velocity at a specific moment. This concept is crucial when approximating values such as the final velocity at impact, by refining the intervals over which the average rate of change is measured.

Position Function

A position function represents the location of an object as a function of time, typically incorporating factors such as initial position, velocity, and acceleration. It is fundamental in kinematics for describing the motion of objects under various forces.

Quadratic Equation

Often arising in problems of projectile motion under constant acceleration, quadratic equations involve terms up to the second degree. Solving these equations allows for the determination of key points in motion, such as when an object reaches the ground.

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