Law of Conservation of Energy: Definition, Formula, Derivation (w/ Examples)

Because physics is the study of how matter and energy flow, the ​law of conservation of energy​ is a key idea to explaining everything a physicist studies, and the manner in which he or she goes about studying it.

Physics is not about memorizing units or equations, but about a framework that governs how all particles behave, even if the similarities are not evident at a glance.

The first law of thermodynamics​ is a restatement of this energy conservation law in terms of heat energy: The ​internal energy​ of a system must equal the total of all the work done on the system, plus or minus the heat flowing into or out of the system.

Another well-known conservation principle in physics is the law of the conservation of mass; as you’ll discover, these two conservation laws – and you'll be introduced to two others here as well – are more closely related than meets the eye (or brain).

Newton's Laws of Motion

Any study of universal physical principles should be backed by a review of the three basic laws of motion, hammered into form by Isaac Newton hundreds of years ago. These are:

  • First law of motion (law of inertia):​ An object with constant velocity (or at rest, where v = 0) remains in this state unless an unbalanced external force acts to perturb it. 
  • Second law of motion:​ A net force (Fnet) acts to accelerate objects with mass (m). Acceleration (a) is the rate of change of velocity (v).
  • Third law of motion:​ For every force in nature, there exists a force equal in magnitude and opposite in direction.

Conserved Quantities in Physics

The laws of conservation in physics apply to mathematical perfection in only truly isolated systems. In everyday life, such scenarios are rare. Four conserved quantities are ​mass​, ​energy​, ​momentum​ and ​angular momentum​. The last three of these fall under the purview of mechanics.

Mass​ is just the amount of matter of something, and when multiplied by the local acceleration due to gravity, the result is weight. Mass can no more be destroyed or created from scratch than energy can.

Momentum​ is the product of an object's mass and its velocity (m·​v​). In a system of two or more colliding particles, the total the momentum of the system (the sum of the individual momenta of the objects) never changes as long as there are no frictional losses or interactions with external bodies.

Angular momentum​ (​L​) is just the momentum about an axis of a rotating object, and is equal to m·​v·r​, where r is the distance from the object to the axis of rotation.

Energy​ appears in many forms, some more useful than others. Heat, the form in which is where all energy is ultimately destined to exist, is the least useful in terms of putting it to useful work, and is usually a product.

The law of conservation of energy may be written:


where KE = ​kinetic energy​ = (1/2)m​v2​, PE = ​potential energy​ (equal to m​g​h when gravity is the only force acting, but seen in other forms), IE = internal energy, and E = total energy = a constant.

  • Isolated systems can have mechanical energy converted to heat energy within their boundaries; you can define a "system" to be any setup you choose, as long as you can be certain of its physical characteristics. This this does not violate the conservation of energy law.

Energy Transformations and Forms of Energy

All the energy in the universe arose from the Big Bang, and that total amount of energy cannot change. Instead, we observe energy changing forms continually, from kinetic energy (energy of motion) to heat energy, from chemical energy to electrical energy, from gravitational potential energy to mechanical energy and so on.

Examples of Energy Transfer

Heat is a special type of energy (​thermal energy​) in that, as noted, it is less useful to humans than other forms.

This means that once part of the energy of a system is transformed to heat, it cannot be as easily returned to a more useful form without the input of additional work, which takes additional energy.

The ferocious amount of radiant energy the sun puts out every second and can never in any way reclaim or reuse is a standing testament to this reality, which is continually unfolding all over the galaxy and the universe as a whole. Some of this energy is "captured" in biological processes on Earth, including photosynthesis in plants, which make their own food as well as providing food (energy) for animals and bacteria, and so on.

It can also be captured by products of human engineering, such as solar cells.

Tracking Energy Conservation

High-school physics students typically use pie charts or bar graphs to show the total energy of the system under study and to track its changes.

Because the total amount of energy in the pie (or the sum of the heights of the bars) cannot change, the difference in slices or bar categories demonstrates how much of the total energy at any given point is one form of energy or another.

In a scenario, different charts may be shown at different points to track these changes. For example, note that the amount of thermal energy almost always increases, representing waste in most cases.

For example, if you throw a ball at a 45-degree angle, initially all of its energy is kinetic (because h = 0), and then at the point at which the ball reaches its highest point, its potential energy as a share of total energy is highest.

Both as it rises and as it subsequently falls, some of its energy is transformed into heat as a result of frictional forces from the air, so KE + PE does not remain constant throughout this scenario, but instead decreases while total energy E still remains constant.

(Insert some example diagrams with pie/bar charts tracking energy changes

Kinematics Example: Free Fall

If you hold a 1.5-kg bowling ball from a rooftop 100 m (about 30 stories) above the ground, you can calculate its potential energy given that the value of ​g = 9.8 m/s2​ and PE = m​g​h:

(1.5\text{ kg})(100\text{ m})(9.8\text{ m/s}^2) = 1,470\text{ Joules (J)}

If you release the ball, its zero kinetic energy increases more and more quickly as the ball falls and accelerates. At the instant it reaches the ground, KE must be equal to the value of PE at the beginning of the problem, or 1,470 J. At this moment,


Assuming no energy loss due to friction, conservation of mechanical energy allows you to calculate ​v​, which turns out to be ​44.3 m/s.

What About Einstein?

Physics students might be confused by the famous ​mass-energy​ ​equation​ (​E = mc2​), wondering if it defies the law of ​conservation of energy​ (or ​conservation of mass​), since it implies mass can be converted to energy and vice versa.

It doesn't actually violate either law because it demonstrates that mass and energy are actually different forms of the same thing. It is kind of like measuring them in different units given the different demands of classical and quantum mechanics situations.

In the heat death of the universe, per the third law of thermodynamics, all matter will have been converted into thermal energy. Once this energy conversion is complete, no more transformations can occur, at least not without another hypothetical singular event such as the Big Bang.

The Perpetual Motion Machine?

A "perpetual motion machine" (e.g., a pendulum that swings with the same timing and sweep without ever slowing down) on Earth is impossible because of air resistance and associated energy losses. To keep the gizmo going would require an input of external work at some point, thus defeating the purpose.

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