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 ball is thrown upward from the top of a 96 -foot-high tower with an initial velocity of 80 feet per second. When does the ball reach its maximum height and how high is it at that time?

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 ball is thrown upward from the top of a 96 -foot-high tower with an initial velocity of 80 feet per second. When does the ball reach its maximum height and how high is it at that time?

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Key Concepts

Projectile Motion

Projectile motion describes the motion of an object that is projected into the air and is subject only to the force of gravity. It involves understanding how initial velocity and height, along with gravitational acceleration, influence the trajectory of the object, resulting in a symmetric parabolic path when air resistance is negligible.

Quadratic Functions in Kinematics

Quadratic functions are commonly used in kinematics to model the vertical motion of objects under gravity. The standard equation h = -16t^2 + v?t + h? represents how an object's height changes over time, incorporating the effects of initial velocity, initial height, and the constant acceleration due to gravity, which creates a parabolic curve.

Vertex of a Parabola

The vertex of a parabola in a quadratic function is the point where the maximum or minimum value occurs. In the context of projectile motion, the vertex represents the highest point reached by the object. Finding the vertex, either through the vertex formula or by setting the derivative to zero, provides the time at which the maximum height occurs and the corresponding height.

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