Question

Imagine a book that is falling from a shelf. At a particular moment during its fall, the book has a kinetic energy of $13\mathrm{J}$ and a potential energy with respect to the floor of $72\mathrm{J}$. How does the book's kinetic energy and its potential energy change as it continues to fall? What is its total kinetic energy at the instant just before it strikes the floor? [Section 5.1]

Short answer

As the book falls, its potential energy decreases while its kinetic energy increases. However, due to the conservation of mechanical energy principle, the sum of kinetic and potential energy remains constant at $85J$. Just before the book hits the floor, its potential energy becomes zero, and all of the mechanical energy is in the form of kinetic energy. Therefore, the total kinetic energy just before the book strikes the floor is $85J$.

Step-by-step solution

Understand the conservation of mechanical energy principle

The conservation of mechanical energy principle states that, in the absence of non-conservative forces (such as air resistance), the total mechanical energy (the sum of kinetic and potential energy) of an object remains constant. In this case, the only forces acting on the book are gravity (a conservative force) and a negligible amount of air resistance. Therefore, we can apply the conservation of mechanical energy principle to this problem.

Determine the initial total mechanical energy of the book

The given kinetic energy at the particular moment during the book's fall is $13J$, and its potential energy with respect to the floor is $72J$. The total mechanical energy is the sum of these two values: Total mechanical energy = Kinetic energy + Potential energy Total mechanical energy = $13J+72J=85J$

Analyze how the energies change during the fall

As the book falls, its potential energy decreases due to the book getting closer to the floor. At the same time, its kinetic energy increases because the book accelerates due to gravity. However, the sum of the kinetic and potential energy remains constant at $85J$ throughout the fall.

Calculate the total kinetic energy just before the book hits the floor

Just before the book strikes the floor, its potential energy with respect to the floor becomes zero. The book's total mechanical energy remains constant at $85J$, so all of this energy is now in the form of kinetic energy. Thus, the total kinetic energy just before the book hits the floor is $85J$.