Problem 24 - Chemistry and Chemical Reactivity

Question

A book is held at a height above the floor. It has a certain amount of potential energy. When the book is released, its potential energy is converted to kinetic energy as it falls to the floor. The book hits the floor and comes to rest. According to the law of conservation of energy the amount of energy in the universe is constant. Where has the energy gone?

Step-by-Step Solution

Answer

The energy is converted to sound and thermal energy on impact.

Step 1: Understand Potential Energy

Potential energy is the energy that an object possesses due to its position relative to a reference point. For the book above the floor, its potential energy is calculated as \(PE = mgh\), where \(m\) is mass, \(g\) is gravitational acceleration, and \(h\) is the height above the floor.

Step 2: Analyze the Conversion to Kinetic Energy

When the book is released, it falls under the influence of gravity. During the fall, the potential energy is converted to kinetic energy \(KE\), which is given by \(KE = \frac{1}{2}mv^2\), where \(v\) is the velocity of the book. Right before it hits the floor, all the potential energy has been converted to kinetic energy.

Step 3: Consider Impact with the Floor

When the book hits the floor and comes to rest, its kinetic energy is transferred to other forms. The energy is not lost but transformed. Mainly, it is converted into sound energy (the noise of the impact) and thermal energy, slightly warming the book and floor.

Step 4: Apply the Law of Conservation of Energy

The law of conservation of energy states that energy cannot be created or destroyed; it can only be transformed. In this case, the initial potential energy is transformed into kinetic energy during the fall and then into sound and thermal energy upon impact, illustrating that the total energy remains constant.

Key Concepts

Potential Energy

When an object is positioned at a certain height above the ground, it possesses what's known as potential energy. This form of energy is harbored in the object due to its elevated position relative to a reference point. Consider it like energy that's waiting to be used. For our book, its potential energy—when held at a height—is determined using the formula: \[ PE = mgh \] where:

• \( m \) is the mass of the book
• \( g \) stands for the acceleration due to gravity
• \( h \) symbolizes the height of the book above the floor

The higher the book is lifted, the greater its potential energy. This energy is like storing up potential activity, ready to be transformed when the conditions change.

Kinetic Energy

As soon as the book is released, it starts to fall, and the energy that was once potential begins its transformation into kinetic energy. Kinetic energy is the energy of motion, harnessed by objects in movement. For our falling book, this transformation is governed by the formula: \[ KE = \frac{1}{2}mv^2 \] where:

• \( m \) is the mass of the book
• \( v \) represents the velocity of the book as it falls

When the book speeds downward, its velocity increases, and hence, its kinetic energy surges. By the time the book is just about to touch the ground, all its potential energy has been fully converted into kinetic energy. This transformation of energy highlights how energy changes form but never disappears.

Energy Transformation

Upon hitting the floor, the book's journey of energy transformation continues. Even though the book stops moving and comes to rest, energy doesn't just vanish. This is thanks to the law of conservation of energy, which asserts that energy can only change forms—not be created or destroyed. Here's how the energy transforms at this stage:

• Sound Energy: The impact generates a sound, transferring some energy as audible noise.
• Thermal Energy: A slight amount of energy turns into heat due to friction, warming the book and the floor ever so slightly.

Effectively, the energy is transferred to the environment in various forms, keeping the total amount constant. This seamless transition from potential to kinetic, and finally to other energy forms upon impact, exemplifies the intricate balance and ongoing exchange in energy transformation. Thus, even when not visible, energy remains active in different guises.