An object thrown directly upward is at a height of s=-16...

An object thrown directly upward is at a height of $s=-16 t^{2}+48 t+256$ feet after $t$ seconds (see Example 4). (a) What is its initial velocity? (b) When does it reach its maximum height? (c) What is its maximum height? (d) When does it hit the ground? (e) With what speed does it hit the ground?

An object thrown directly upward is at a height of $s=-16 t^{2}+48 t+256$ feet after $t$ seconds (see Example 4).
(a) What is its initial velocity?
(b) When does it reach its maximum height?
(c) What is its maximum height?
(d) When does it hit the ground?
(e) With what speed does it hit the ground?

Key Concepts

Projectile Motion

This concept involves the study of objects that are thrown into the air and move under the influence of gravity, assuming no air resistance. It explains how the object’s path is determined by its initial velocity and the constant acceleration due to gravity, resulting in a parabolic trajectory.

Kinematic Equations

These equations describe the motion of an object under constant acceleration. They relate displacement, initial velocity, time, and acceleration, forming the foundation for solving problems involving vertical motion, such as computing heights, velocities, and durations of flight.

Differentiation in Kinematics

Differentiation is used in kinematics to determine the instantaneous velocity of an object by taking the derivative of its displacement function with respect to time. This technique helps in finding initial velocities and understanding how the velocity changes over time.

Quadratic Functions and Their Vertex

When the displacement of a projectile is expressed as a quadratic function of time, the vertex of the parabola represents the maximum or minimum point of the trajectory. The vertex formula is used to pinpoint the time at which the maximum height is reached, along with computing the highest point attained.

Solving Quadratic Equations

This process involves finding the roots of a quadratic function by setting the displacement equal to a particular value, such as zero for ground impact. Techniques such as factoring, completing the square, or using the quadratic formula are employed to determine the times at which specific events occur in projectile motion.

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