A stone is dropped from the top of a tower and the location...

A stone is dropped from the top of a tower and the location from the starting point $s$ (in feet) of the stone at time $t$ (in seconds) is given by the equation $s(t)=-16 t^2$, where direction upward is considered positive. If the building is 256 feet tall, find (a) the velocity and (b) the acceleration of the stone after 2 seconds.

Key Concepts

Constant Acceleration due to Gravity

In the context of free-fall motion near Earth's surface, the acceleration is constant, typically approximated as -32 feet per second squared in a system where upward is positive. This constant acceleration simplifies the analysis of motion and is reflected in the quadratic form of the position function, where the coefficient relates to one half of the gravitational acceleration.

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without referring to the forces that cause the motion. In problems involving free fall or projectile motion, kinematic equations describe the relationship between displacement, velocity, acceleration, and time, providing a framework to analyze an object’s motion under constant acceleration.

Differentiation in Motion Analysis

Differentiation is an essential mathematical tool in kinematics used to determine velocity and acceleration from a position-time function. Taking the first derivative of the position function with respect to time yields the velocity function, which describes the rate of change of position, while the second derivative provides the acceleration function, indicating the rate at which the velocity changes with time.

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