Conservation of energy

Conservation of energy (the First Law of Thermodynamics) is a simple law stating that although energy may change form, it cannot disappear altogether, nor can it be created.

More recently, this law has been modified to take into account the equivalence of mass and energy ($$E = mc^2$$) discovered by Einstein. This new law is called the law of conservation of mass and energy. It states that although mass may convert to energy, and vice versa, neither may disappear without compensation in the other quantity. However, the original form of the law is adequate in most everyday situations.

A concrete example of energy conservation is found with falling objects. Near the surface of the Earth, the gravitational potential energy of an object is given by $$mgh$$, where m is the object's mass, g is the acceleration due to gravity at that place, and h is the object's height from where the object started to where it is now (measure with downward values negative). When an object falls, its potential energy is converted into kinetic energy (given by $$(1/2)mv^2$$). In math:

$$-mgh = \frac{1}{2}mv^2 $$

where m is mass, v is velocity. There is a negative sign in front of the potential energy because when the object is gaining kinetic energy, it will be losing potential energy.

Another, more useful way of interpreting conservation of energy is by using the following formula

$$K_1+U_1+W other=K_2+U_2$$