An orange falling from 20 feet has a height of s(t)= 20-16...

An orange falling from 20 feet has a height of $s(t)=$ $20-16 t^{2}$ 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 ground. (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. (c) From the data in your table, make a guess for the instantaneous final velocity of the orange at the moment it hits the ground.

An orange falling from 20 feet has a height of $s(t)=$ $20-16 t^{2}$ 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 ground.
(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.
(c) From the data in your table, make a guess for the instantaneous final velocity of the orange at the moment it hits the ground.

Key Concepts

Average Rate of Change

The average rate of change of a function over an interval is calculated by dividing the change in the function’s value by the change in the independent variable. In the context of motion, it represents the object's average velocity over a specific time interval and is determined by considering the difference in position over that interval.

Position Function

A position function is a mathematical expression that describes the location of an object as a function of time. In the context of motion problems, it is used to model how the object's height or position changes over time, allowing one to analyze various aspects of the object's movement, such as when it reaches a particular point.

Solving for Zeros of a Function

Finding the zeros of a function involves solving for the points in time when the function's output is zero. In motion problems, determining when the position function equals zero helps identify when an object, such as a falling object, reaches the ground. This often requires solving algebraic equations using techniques like factoring or applying the quadratic formula.

Graphing a Function

Graphing a function is a visual method to represent the relationship between variables, such as time and position. It helps in understanding the behavior of the function, identifying key features like intercepts and the general trend, and providing an intuitive sense of how an object moves over time.

Instantaneous Rate of Change

The instantaneous rate of change is the limit of the average rate of change as the time interval approaches zero. This concept, which is central to differential calculus, gives the exact velocity of the object at a specific moment, such as the moment it hits the ground.

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