A ball is thrown vertically upward from the ground. If its...

A ball is thrown vertically upward from the ground. If its height $h$ measured in feet is given in terms of time $t,$ measured in seconds by the equation $h=-16 t^{2}+96 t,$ determine (a) how high the ball will go and how long does it takes to reach that height, (b) how long does it take for the ball to come back to the ground?

Key Concepts

Solving Quadratic Equations

Solving quadratic equations is a method used to find the values of the variable that satisfy the equation. In projectile problems, solving the quadratic equation helps determine when the moving object reaches a specific height, such as when it returns to the ground (when the height is zero). Techniques such as factoring, completing the square, or using the quadratic formula are commonly applied.

Projectile Motion

Projectile motion refers to the motion of an object that is projected into the air and is influenced only by the force of gravity (and, in many simplified problems, neglecting air resistance). This motion is typically modeled by quadratic functions, which enable the calculation of critical points such as peak height, ascent time, and total time of flight before returning to the ground.

Vertex of a Parabola

The vertex of a parabola is the point at which the function reaches its extreme value, either maximum or minimum. In the context of projectile motion, the vertex represents the highest point reached by the object. Finding the vertex allows us to determine the maximum height and the time at which this height is achieved.

Quadratic Function

A quadratic function is a second degree polynomial that is often represented in the form ax² + bx + c. It is fundamental in modeling trajectories and various natural phenomena, as it forms a parabolic curve when graphed. This concept is crucial in understanding how the height of a projectile changes over time under the influence of constant acceleration due to gravity.

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