A stone thrown upward from the top of a 320 -foot cliff at...

A stone thrown upward from the top of a 320 -foot cliff at $128 \mathrm{ft} / \mathrm{sec}$ eventually falls to the beach below. (a) How long does the stone take to reach its highest point? (b) What is its maximum height? (c) How long before the stone hits the beach? (d) What is the velocity of the stone on impact?

A stone thrown upward from the top of a 320 -foot cliff at $128 \mathrm{ft} / \mathrm{sec}$ eventually falls to the beach below.
(a) How long does the stone take to reach its highest point?
(b) What is its maximum height?
(c) How long before the stone hits the beach?
(d) What is the velocity of the stone on impact?

Key Concepts

Kinematic Equations

These equations describe the relationship among displacement, initial velocity, acceleration, and time during uniformly accelerated motion. They are essential for solving problems related to projectile motion, such as determining the time to reach maximum height, the maximum height itself, and the total time of flight until impact.

Solving Quadratic Equations in Kinematics

Many kinematics problems result in quadratic equations when solving for time or displacement, especially when the object's motion includes both upward and downward travel. The ability to solve these quadratic equations accurately, often using the quadratic formula, is crucial for determining the times at which key events (like impact with the ground) occur.

Uniformly Accelerated Motion

This concept refers to the motion of an object under a constant acceleration, such as gravity. In problems like these, acceleration is constant (for example, gravity at 32 ft/s² downward), which simplifies the analysis of the object's motion by allowing the use of standard kinematic equations.

Determining Maximum Height

In projectile motion, the maximum height is reached when the vertical velocity becomes zero. By setting the velocity equation to zero and solving for time, one can then substitute that time back into the displacement equation to find the maximum height. This technique is fundamental in analyzing upward trajectories.

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