A ball is thrown in the air from the top of a building. Its...

A ball is thrown in the air from the top of a building. Its height, in meters above ground, as a function of time, in seconds, is is iven by $h(t)=-4.9 t^{2}+24 t+8 .$ How long does it take to reach maximum height?

Key Concepts

Quadratic Function

A quadratic function is a second-degree polynomial generally expressed in the form f(t) = at² + bt + c. Its graph is a parabola, which opens upward if a is positive and downward if a is negative. This characteristic makes quadratic functions suitable for modeling scenarios like projectile motion where acceleration is constant, as the motion produces a parabolic trajectory.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on its graph, depending on whether it opens downward or upward, respectively. For quadratic functions, the vertex represents the maximum or minimum value of the function. It can be found using the formula t = -b/(2a), which provides the time when the maximum or minimum occurs. In the context of projectile motion, when the quadratic function represents height, the vertex gives the time at which the object reaches its maximum height.

Projectile Motion

Projectile motion is the motion of an object thrown or projected into the air, subject only to acceleration due to gravity. This type of motion can be represented by a quadratic function, where the height of the projectile as a function of time follows a parabolic path. Understanding projectile motion involves analyzing this quadratic model to determine key features such as maximum height and time of flight, with the vertex of the parabola indicating the point of maximum height.

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