Use the position equation given in Applications as the model...

Use the position equation given in Applications as the model for the problem. At 2080 feet tall, the Tokyo Skytree in Tokyo, Japan, is the world's tallest tower. An object is dropped from the top of the tower. a. Find the position equation $s=-16 t^{2}+v_{0} t+s_{0}$. b. Complete the table. c. From the table in part (b), determine the time interval during which the object reaches the ground. Find the time algebraically.

Use the position equation given in Applications as the model for the problem.
At 2080 feet tall, the Tokyo Skytree in Tokyo, Japan, is the world's tallest tower. An object is dropped from the top of the tower.
a. Find the position equation $s=-16 t^{2}+v_{0} t+s_{0}$.
b. Complete the table.
c. From the table in part (b), determine the time interval during which the object reaches the ground. Find the time algebraically.

Key Concepts

Kinematic Equations

This concept involves understanding the fundamental equation for motion under constant acceleration: s = s0 + v0t + (1/2)at². It models the position of an object over time when initial conditions, like the starting position and velocity, and a constant acceleration (for example, gravity) are known.

Free-Fall Conditions

When an object is dropped, it is released from rest, meaning its initial velocity (v0) is zero. The only acceleration acting on it is due to gravity, which in these problems is often taken as -32 ft/s², resulting in a simplification of the kinematic equation where the term involving the initial velocity drops out.

Quadratic Equations in Motion

Setting the position equation equal to zero to determine when the object touches the ground leads to a quadratic equation in the variable time (t). Solving this quadratic equation, whether by factoring, completing the square, or applying the quadratic formula, is essential for finding the exact time of impact.

Tabular Analysis of Motion

Creating a table to calculate the position of the object at various time increments helps in visualizing the trajectory and approximating the time interval during which the object reaches the ground. This method serves both as a verification of the algebraic solution and as a practical tool to understand the motion's progression.

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