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

Use the position function
$$
s(t)=-16 t^{2}+v_{0} t+s_{0}
$$
to solve this problem. A projectile is fi red vertically upward from ground level with an initial velocity of 48 feet per second. During which time period will the projectile’s height exceed 32 feet?

Key Concepts

Quadratic Equations and Inequalities

Solving quadratic equations is critical for finding the time instances when the projectile reaches a specific height. Moreover, setting up and solving quadratic inequalities helps determine the time duration over which the projectile's height exceeds a given value, by finding the roots and analyzing the interval between them.

Projectile Motion Kinematics

This concept involves studying objects moving under constant acceleration, particularly due to gravity. In vertical projectile motion, the object's height is represented as a function of time using a quadratic equation which takes into account the initial velocity, initial height, and the acceleration due to gravity.

Position Function

The position function, typically given in the form s(t) = -½gt² + v?t + s?, models the height of a projectile at any time t. This quadratic function captures the upward and downward movement of the projectile, indicating when it will rise above or fall below certain positions.

Gravitational Acceleration

In the context of projectile motion on Earth, gravitational acceleration is a constant that influences the object's motion. In the imperial system, it is commonly represented as 32 feet per second squared, and its effect is incorporated into the position function as a term that is halved (resulting in -16t²) to account for the integration of acceleration over time.

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