The law of conservation of energy is a physical law that states that the total energy of an isolated system is a constant, although energy can change forms. In other words, energy is conserved over time. The law of conservation of energy is the first law of thermodynamics. French mathematician and philosopher Émilie du Châtelet first proposed and tested the law in the 18th century.

### Formulas for the Law of Conservation of Energy

There are a few different ways of writing the formula for the law of conservation of energy. One of the most common formulas describes the relationship between kinetic energy (K) and potential energy (U):

**K _{1} + U_{1} = K_{2} + U_{2}**

In this case, the total energy of the system is a constant, but energy converts between potential and kinetic energy.

For calculations involving frictionless carts, swings, pendulums, throwing a ball, etc., there is another useful form of the equation for the conservation of energy, which uses the following formulas for potential and kinetic energy:

U = mgh ; where U is potential energy, m is mass, g is acceleration due to gravity, and h is height

K = ½mv^{2} ; where m is mass and v is velocity

Total energy is the sum of potential and kinetic energy:

**E _{total} =**

**mgh + ½mv**

^{2}This formula works well for physics problems where there is no friction. More complex equation cover the situation where some energy gets converted into heat via friction.

**Conservation of Energy Example Problem**

See the formula for conservation of energy in action with this common physics problem involving a cart traveling on a frictionless track.

Another form of the laws of conservation of energy states that the internal energy (∆E) of a system is the sum of the heat flow (Q) across its boundaries (q) and the work done on the system (W).

**∆E = Q + W**

### Examples of the Law of Conservation of Energy

There are many examples of the law of conservation of energy in everyday life:

- The energy of a child on a swing changes between potential and kinetic energy. At the top of the swing, all of the energy is potential. At the bottom of the swing, it’s all kinetic. The energy is a mixture of kinetic and potential energy between these two points. In a frictionless system, the potential energy at the top equals the kinetic energy at the bottom, which equals the sum of the kinetic and potential energy at the other points.
- A swinging pendulum also illustrates a conversion between kinetic and potential energy, exactly like a swing. Of course, in both the swing and pendulum examples, friction plays a role. The conserved energy really is a mixture of kinetic energy, potential energy, and thermal energy or heat.
- A car converts chemical energy (gasoline) into kinetic energy. Here again, so energy is lost as heat, but the sum of the forms of energy remains constant.
- As an apple falls from a tree, it starts out with potential energy. As it falls, it has a mixture of kinetic and potential energy. In the instant it strikes the ground, all of its energy is kinetic. The sum of its potential and kinetic energy is a constant value.
- A flashlight converts chemical energy from its battery into electrical energy, which is then converted into light and heat.
- A speaker converts electrical energy into sound, which is another form of energy.
- Generators convert mechanical energy into electrical energy.
- Your body converts chemical energy from food into mechanical energy (moving muscles), different chemical energy molecules, and heat.
- An exploding firework converts chemical potential energy into kinetic energy, light, heat, and sound.

### Classical Mechanics vs General Relativity

In classical mechanics, the law of conservation of energy and the law of conservation of mass are two separate laws. However, they combine in relativity in Einstein’s famous equation:

E = mc^{2}

This equation shows mass can convert into energy, and vice versa. The law of conservation of energy still holds true, as long as the reference from of the observer remains unchanged.

### Perpetual Motion Machines

One consequence of the law of conservation of energy is that is means perpetual motion machines of the first kind are impossible. These are machines that do work forever without any additional energy input. While perpetual motion that does work might look good on paper, it doesn’t work in the real world because some energy in a machine changes form into heat. Usually, this is from friction. So, to keep a machine running actually requires a continuous input of energy.

### Exceptions

Remember, the law of conservation of energy applies to a closed system. Sometimes it isn’t easy or even possible to define or isolate a system. This comes into play in general relativity, where systems don’t always have time translation symmetry. For example, conservation of energy isn’t necessarily defined for curved spacetime or time crystals.

### References

- Feynman, Richard (1970).
*The Feynman Lectures on Physics Vol I*. Addison Wesley. ISBN 978-0-201-02115-8. - Gibney, Elizabeth (2017). “The quest to crystallize time”.
*Nature*. 543 (7644): 164–166. doi:10.1038/543164a - Hagengruber, Ruth (ed.) (2011).
*Émilie du Chatelet: Between Leibniz and Newton*. Springer. ISBN 978-94-007-2074-9. - Kroemer, Herbert; Kittel, Charles (1980).
*Thermal Physics*(2nd ed.). W. H. Freeman Company. ISBN 978-0-7167-1088-2. - Serway, Raymond A.; Jewett, John W. (2004).
*Physics for Scientists and Engineers*(6th ed.). Brooks/Cole. ISBN 978-0-534-40842-8.