Final answer:
The total energy of a system, combining kinetic and potential energy, remains constant in the absence of non-conservative forces. A falling ball transforms its potential energy into kinetic energy as it falls, but the sum of both types of energy is conserved, assuming no energy is lost. This conservation is represented by the equation E = K + U.
Step-by-step explanation:
Conservation of Energy in a Falling Ball
The total energy of a system is the sum of its kinetic energy (K) and potential energy (U). In the case of a falling ball, when the ball is held at a specific height, it possesses gravitational potential energy. As the ball is dropped, its potential energy is converted into kinetic energy. However, the sum of kinetic and potential energy remains constant during the fall, assuming no energy is lost to air resistance or other non-conservative forces. This is energy conservation in action: E = K + U remains constant.
For a ball dropped from a height and allowed to bounce, the following energy transformations occur: On impact, some of the kinetic energy is transferred into other forms such as sound and heat, and if the ball bounces to half of its original height, this indicates that half of the initial potential energy has been transferred to these non-mechanical forms. The remaining potential energy is converted back into kinetic energy as the ball rises. This cycle demonstrates how energy is conserved within a given system and transformed between different types.
If we were to graph the total momentum versus time for two carts colliding with conservation of momentum, we would see a constant value since momentum is conserved. On a graph of total kinetic energy versus time, assuming no external forces do work on the system, we would see that the kinetic energy remains constant throughout the interaction, showcasing energy conservation. However, if we consider the bounce of a ball, energy is lost to the environment - displayed as a decrease in kinetic energy after each bounce.
In summary, the Principle of Conservation of Mechanical Energy states that in an isolated system with only conservative forces acting, the total mechanical energy remains constant.