A ball is thrown straight up from the top of a building 144...

A ball is thrown straight up from the top of a building 144 ft. tall with an initial velocity of $64 \mathrm{ft}$ per second. The height $s(t)$ (in feet) of the ball from the ground, at time $t$ (in seconds), is given by $s(t)=144+64 t-16 t^{2} .$ Find the maximum height attained by the ball.

Key Concepts

Quadratic Functions

A quadratic function is a polynomial of degree two that is typically expressed in the form ax² + bx + c. It graphically represents a parabola that opens upward if the coefficient a is positive or downward if it is negative. Quadratic functions are fundamental in modeling situations where acceleration is constant, such as objects moving under the influence of gravity, making them essential in physics for analyzing projectile motion.

Vertex of a Parabola

The vertex of a parabola is the point at which the function reaches its maximum (if the parabola opens downward) or minimum (if it opens upward). For quadratic functions, the vertex can be found using the formula -b/(2a) for the x-coordinate, with the corresponding y-coordinate determined by substituting this value back into the function. This concept is crucial for determining the peak or extreme value of a quadratic model, such as the maximum height reached by an object in projectile motion.

Projectile Motion

Projectile motion refers to the motion of an object that is thrown or projected into the air and subject only to the acceleration due to gravity. The trajectory of the object typically follows a parabolic path, and the analysis involves understanding how initial velocity, gravitational acceleration, and initial position affect the object's path. It is a key concept in classical mechanics and is often modeled using quadratic equations to predict the maximum height, range, and time of flight of the object.

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