The law of conservation of energy states that the total energy of a closed system remains constant over time. This means that the total energy before a process or event should be equal to the total energy after the process or event.
In the case of an object falling in a gravitational field, if we consider only the gravitational potential energy (PE = mgh) and the kinetic energy (KE = 1/2mv^2), the law of conservation of energy would indeed suggest that the sum of these two energies before the fall should be equal to the sum after the fall.
However, it is important to consider the broader context and take into account other factors that may be involved in the system. When an object falls, it experiences various forms of energy transfer and conversion.
During free fall, in the absence of other forces such as air resistance, the object's potential energy is converted into kinetic energy. As the object falls, its potential energy decreases, while its kinetic energy increases proportionally. The law of conservation of energy is still satisfied because the total mechanical energy (the sum of potential and kinetic energy) remains constant.
However, if we consider additional factors such as air resistance, then energy dissipation occurs due to the work done against air resistance. In this case, the object's mechanical energy is not conserved, as some of it is transformed into other forms of energy, such as heat or sound.
In reality, in most scenarios involving falling objects on Earth, air resistance is present, and it can significantly affect the motion and energy dynamics. As a result, the calculated potential energy (mgh) and the calculated kinetic energy (1/2mv^2) may not be equal due to the dissipation of energy through factors like air resistance.
To accurately account for the energy changes in a falling object, a more comprehensive analysis that incorporates all relevant forces and factors, including air resistance, is required.