Understanding the Concepts
* Potential Energy: An object at a height has potential energy due to its position in a gravitational field. This energy is given by: PE = mgh, where:
* m = mass (kg)
* g = acceleration due to gravity (approximately 9.8 m/s²)
* h = height (m)
* Kinetic Energy: An object in motion has kinetic energy. This energy is given by: KE = (1/2)mv², where:
* m = mass (kg)
* v = velocity (m/s)
* Conservation of Energy: In this scenario, the book's potential energy at the top of the shelf is converted into kinetic energy as it falls.
Solving the Problem
1. Find the kinetic energy at impact:
KE = (1/2)mv² = (1/2) * 5 kg * (5.7 m/s)² = 81.225 J (Joules)
2. The kinetic energy at impact is equal to the potential energy at the top:
PE = KE
mgh = 81.225 J
3. Solve for the height (h):
h = KE / (mg) = 81.225 J / (5 kg * 9.8 m/s²) = 1.66 m
Therefore, the book fell from a height of approximately 1.66 meters.
