Use the formula for the height h of an object that is...

Use the formula for the height h of an object that is traveling vertically (subject only to gravity) at time $t$ : $$h=-16 t^{2}+v_{0} t+h_{0}$$ where $h_{0}$ is the initial height and $v_{0}$ is the initial velocity; $t$ is measured in seconds and h in feet. A bullet is fired upward from ground level with an initial velocity of 1800 feet per second. How high does it go?

Use the formula for the height h of an object that is traveling vertically (subject only to gravity) at time $t$ : $$h=-16 t^{2}+v_{0} t+h_{0}$$
where $h_{0}$ is the initial height and $v_{0}$ is the initial velocity; $t$ is measured in seconds and h in feet.
A bullet is fired upward from ground level with an initial velocity of 1800 feet per second. How high does it go?

Key Concepts

Maximizing a Quadratic Function

Since the height equation is a quadratic function in time, its maximum point (vertex) represents the maximum height reached by the object. Finding the vertex, by either completing the square or using the formula t = -b/(2a), is a standard technique for determining the peak of the object's trajectory.

Acceleration due to Gravity in Imperial Units

In the given equation, the term -16t² arises from the acceleration due to gravity in feet per second squared, which is approximately -32 ft/s². The equation is structured with a factor of one half, leading to the coefficient -16 instead of -32, highlighting how gravitational acceleration is integrated into the kinematic formula for vertical motion.

Projectile Motion

Projectile motion refers to the motion of an object thrown or launched into the air, subject only to the acceleration due to gravity. In this context, the formula h = -16t² + v?t + h? represents the vertical component of projectile motion, modeling how an object's height changes over time when air resistance is negligible.

Kinematic Equations

Kinematic equations are mathematical formulations that describe the motion of objects under constant acceleration. The height equation provided is one of these kinematic equations, specifically for vertical motion, which includes the initial velocity, initial height, and the acceleration due to gravity.

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