能量守恒

Conservation of energy states that the net energy of a closed system is fixed. Let's analyze this definition.

Net energy means the sum of all forms of energy in a system. For example, a simple ball being thrown off a cliff has many forms of energy. Most importantly, it has potential energy and kinetic energy. Although, it has other forms of energy like heat energy being transferred to the air through friction and energy on the atomic scale. The net energy of this system is the sum of all of these possible types of energy: $\sum E=E_{potential}+E_{kinetic}+\dots$

To be a closed system, energy can't be escaping the system or else the net energy would obviously be lost (or gained!). For example, the ball is not a closed system because energy is transferred to the air around it in the form of heat.

An invariant is something that always remains the same. For example, if you start with the number $0$ and either add or subtract $2$ each turn, its parity is invariant (parity means if it's even or odd, and the number will always be even).

This means that no matter what operation we do, in any closed system, the net energy of the system will always remain the same.

Problem: A ball drops off a cliff in a vacuum starting at height $h$. What is the speed at which it hits the ground? What is the speed at which it is halfway to the ground?

Solution: The ball has potential and kinetic energy. The ball loses no energy as heat and all other forms of energy are negligible. At the top of the cliff, it has only potential energy because it's not moving initially. Therefore, $\sum E=E_{potential}=mgh$ at the top. At the bottom, it no longer has potential energy because it has no height. But, since it's moving, it has kinetic energy, so $\sum E=E_{kinetic}=\frac12mv^2$ at the bottom. Since $\sum E$ is invariant, $mgh=\frac12mv^2$, so $v^2=2gh$ or $v=\sqrt{2gh}$. Fun fact: the velocity at which it hits the ground doesn't depend on its mass!

For the second part, it has the same initial net energy. However, halfway down the cliff it has both kinetic and potential energy! Here, $\sum E=E_{potential}+E_{kinetic}=mg\frac{h}2+\frac12mv^2$. Since $\sum E$ is invariant, $mgh=mg\frac{h}2+\frac12mv^2$, so at this point $v^2=gh$, so $v=\sqrt{gh}$.

Remarks: Conservation of energy is a very powerful tool!

The law of conservation of energy states that, in a closed system, one where there are no external forces, the total energy of the system is a constant. Namely, there is such number, the total energy, which is unchanged regardless of the internal noise of the system.

Here's a great demo which I just found from PhET Interactive Simulations!

Demo

Just like to add that the conservation of Energy equation is a state function so the process in which the object moves neither affects your calculations nor should be included in them.