Question

a book is falling from the top shelf of a bookcase. which statement correctly describes the change in the energy of the book?

the potential energy and kinetic energy remain the same.

the potential energy decreases as kinetic energy increases.

the kinetic energy decreases as potential energy increases.

the kinetic energy remains at zero, but potential energy increases.

Explanation:

To solve this, we recall the concepts of potential energy (PE) and kinetic energy (KE). Potential energy (due to height) is given by \( PE = mgh \), where \( m \) is mass, \( g \) is gravity, and \( h \) is height. Kinetic energy is \( KE=\frac{1}{2}mv^{2} \), where \( v \) is velocity. As the book falls, its height (\( h \)) decreases, so \( PE \) decreases. At the same time, its velocity (\( v \)) increases (since it accelerates due to gravity), so \( KE \) increases. Energy is conserved (ignoring air resistance), so the decrease in PE is converted into an increase in KE.

Now let's analyze each option:

• "The potential energy and kinetic energy remain the same." → Incorrect, because height and velocity change, so PE and KE must change.
• "The potential energy decreases as kinetic energy increases." → Correct, matches our analysis of PE (decreasing with height) and KE (increasing with velocity) during the fall.
• "The kinetic energy decreases as potential energy increases." → Incorrect, this would be the case if the book were moving upward (gaining height, losing speed), but the book is falling downward.
• "The kinetic energy remains at zero, but potential energy increases." → Incorrect, the book is moving (so KE > 0) and its height is decreasing (so PE is decreasing, not increasing).

Answer:

The potential energy decreases as kinetic energy increases.