2. A $250 \, \text{kg}$ Rock Falls Off A Cliff And Comes To Rest On The Ground, Which Is $40 \, \text{m}$ Below The Cliff.- At What Point Is The Rock's Potential Energy At Maximum?- Where Is The Kinetic Energy
Introduction
When a rock falls off a cliff, it experiences a significant change in its potential energy due to the change in its height. As it falls, its potential energy is converted into kinetic energy, which is the energy of motion. In this article, we will explore the concept of conservation of energy and apply it to a rock's journey from the cliff to the ground.
Conservation of Energy
The law of conservation of energy states that energy cannot be created or destroyed, only converted from one form to another. In the case of the rock, its potential energy is converted into kinetic energy as it falls. The total energy of the rock remains constant throughout its journey.
Potential Energy
The potential energy of an object is given by the formula:
U = mgh
where U is the potential energy, m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object above the ground.
In this case, the rock has a mass of 250 kg and a height of 40 m above the ground. Therefore, its potential energy is:
U = 250 kg x 9.8 m/s^2 x 40 m = 98,000 J
Kinetic Energy
The kinetic energy of an object is given by the formula:
K = (1/2)mv^2
where K is the kinetic energy, m is the mass of the object, and v is its velocity.
As the rock falls, its potential energy is converted into kinetic energy. At the point where the rock's potential energy is at maximum, its kinetic energy is zero. This is because the rock has not yet started to fall, and therefore, it has no velocity.
Maximum Potential Energy
The rock's potential energy is at maximum when it is at its highest point, which is the cliff. At this point, its height is 40 m above the ground, and its potential energy is 98,000 J.
Where is the Kinetic Energy?
As the rock falls, its potential energy is converted into kinetic energy. At the point where the rock's potential energy is at maximum, its kinetic energy is zero. This is because the rock has not yet started to fall, and therefore, it has no velocity.
However, as the rock falls, its kinetic energy increases. At the point where the rock comes to rest on the ground, its kinetic energy is maximum, and its potential energy is zero.
Graphical Representation
The following graph shows the rock's potential energy and kinetic energy as a function of its height.
Potential Energy (U) vs. Height (h)
| Height (h) | Potential Energy (U) |
|---|---|
| 40 m | 98,000 J |
| 30 m | 74,500 J |
| 20 m | 49,000 J |
| 10 m | 24,500 J |
| 0 m | 0 J |
Kinetic Energy (K) vs. Height (h)
| Height (h) | Kinetic Energy (K) |
|---|---|
| 40 m | 0 J |
| 30 m | 12,500 J |
| 20 m | 25,000 J |
| 10 m | 37,500 J |
| 0 m | 49,000 J |
Conclusion
In conclusion, the rock's potential energy is at maximum when it is at its highest point, which is the cliff. At this point, its kinetic energy is zero. As the rock falls, its potential energy is converted into kinetic energy, and at the point where the rock comes to rest on the ground, its kinetic energy is maximum, and its potential energy is zero.
References
- [1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
- [2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.
Discussion
What happens to the rock's energy as it falls? Does it lose energy or gain energy?
The rock's energy is conserved, meaning that it does not lose or gain energy. Instead, its potential energy is converted into kinetic energy as it falls.
What is the relationship between the rock's potential energy and its height?
The rock's potential energy is directly proportional to its height. As the rock's height increases, its potential energy also increases.
What is the relationship between the rock's kinetic energy and its velocity?
The rock's kinetic energy is directly proportional to its velocity. As the rock's velocity increases, its kinetic energy also increases.
Answer Key
- The rock's potential energy is at maximum when it is at its highest point, which is the cliff.
- The rock's kinetic energy is zero when it is at its highest point, which is the cliff.
- The rock's kinetic energy is maximum when it comes to rest on the ground.
- The rock's potential energy is zero when it comes to rest on the ground.
Q&A: Conservation of Energy and the Rock's Journey =====================================================
Q: What is the law of conservation of energy?
A: The law of conservation of energy states that energy cannot be created or destroyed, only converted from one form to another. In the case of the rock, its potential energy is converted into kinetic energy as it falls.
Q: What is potential energy?
A: Potential energy is the energy an object has due to its position or state. In the case of the rock, its potential energy is due to its height above the ground.
Q: What is kinetic energy?
A: Kinetic energy is the energy an object has due to its motion. In the case of the rock, its kinetic energy is due to its velocity as it falls.
Q: At what point is the rock's potential energy at maximum?
A: The rock's potential energy is at maximum when it is at its highest point, which is the cliff.
Q: Where is the rock's kinetic energy at maximum?
A: The rock's kinetic energy is at maximum when it comes to rest on the ground.
Q: What happens to the rock's energy as it falls?
A: The rock's energy is conserved, meaning that it does not lose or gain energy. Instead, its potential energy is converted into kinetic energy as it falls.
Q: What is the relationship between the rock's potential energy and its height?
A: The rock's potential energy is directly proportional to its height. As the rock's height increases, its potential energy also increases.
Q: What is the relationship between the rock's kinetic energy and its velocity?
A: The rock's kinetic energy is directly proportional to its velocity. As the rock's velocity increases, its kinetic energy also increases.
Q: Can the rock's energy be converted back into potential energy?
A: No, the rock's energy cannot be converted back into potential energy. Once the rock's potential energy is converted into kinetic energy, it cannot be converted back into potential energy.
Q: What is the significance of the rock's journey in understanding conservation of energy?
A: The rock's journey is a simple example of how energy is conserved in the universe. It illustrates the concept of potential and kinetic energy and how they are related.
Q: Can you think of other examples of conservation of energy in everyday life?
A: Yes, there are many examples of conservation of energy in everyday life. For example, when you ride a bike, your potential energy is converted into kinetic energy as you pedal. When you throw a ball, its potential energy is converted into kinetic energy as it flies through the air.
Q: What are some real-world applications of conservation of energy?
A: Conservation of energy has many real-world applications, including:
- Designing more efficient engines and machines
- Improving the efficiency of power plants and other energy-producing systems
- Developing new technologies for energy storage and transmission
- Understanding and predicting the behavior of complex systems, such as the Earth's climate and weather patterns.
Conclusion
In conclusion, the rock's journey from the cliff to the ground is a simple example of how energy is conserved in the universe. It illustrates the concept of potential and kinetic energy and how they are related. By understanding conservation of energy, we can design more efficient systems, improve the efficiency of power plants and other energy-producing systems, and develop new technologies for energy storage and transmission.