The law of conservation of energy is one of the basic laws of physics and therefore governs the microscopic motion of individual atoms in a chemical reaction. The law of conservation energy states:

*In a closed system, i.e., a system that isolated from its surroundings, the total energy of the system is conserved.*

In SI units, energy has units of *Joules*.
1 Joule = 1 kgms.

**Some forms of energy**:

- 1.
- Kinetic energy - energy of motion.
- 2.
- Potential energy - energy of ``location'' with respect to some reference point.
- 3.
- Chemical energy - energy stored in chemical bonds, which can be released in reactions.
- 4.
- Electrical energy - energy created by separating charges; energy stored in a battery, for example.
- 5.
- Thermal energy - energy given off as heat, such as friction.

Since everything has a microscopic origin, the last three are really
special cases of potential and kinetic energies, however, the classification
is useful.

The *kinetic energy* of an object of mass ,
moving with a velocity
is given by

Recall that is the speed of the particle, so that the kinetic energy can be written as .

*Potential energy* is a little less straightforward. Since it is
an energy of location with respect to some reference point, the potential
energy of an object depends on the specific situation. An example is
gravity. An object of mass at a height above the Earth's surface
has a gravitational potential energy

where is the acceleration due to the Earth's gravity. In this example, the potential energy has a simple form, namely that it depends linearly on the height , which describes the object's location and may even be varying in time if, for example, the object is actually falling toward the Earth.

In general, the potential energy may not be such a simple function of
location. In this case, one needs a *potential energy curve* to
describe the potential energy as a function of some coordinate describing
the object's location. Consider the example of a mass attached
to a spring moving in one spatial dimension. Let represent the
mass's position along the -axis, and let represent its
equilibrium position. As the mass stretches the spring (),
its potential energy increases, and as it compresses the spring
compresses (), its potential energy increases as well.
The potential energy can be described by a potential energy
function that is symmetric about , as represented
in the figure below:

Notice, also, that the mass actually moves under the action of a force,
which also changes as a function of . In fact, the
force exerted by the spring on the mass can be determined from the
potential energy curve via

That is, the force is the slope of the line tangent to the curve at the point , as shown in the figure above for the point . In terms of a derivative, the force is given by

Note that, at the bottom of the gully, where the curve is flat, the force is 0. The SI units of force are

Consider the case of a diatomic molecule:

The distance between the atoms A and B is . The chemical forces that hold the molecule together come from a potential energy curve that depends only on the distance between them. Such a curve might look like:

The presence of a deep ``well'' indicates a particular distance of lowest potential energy. This corresponds to the bond length of the molecule and is, therefore, the most likely value of the separation of the two atoms (think of a ball placed in a gully of this shape - if placed at the bottom of the well, the force on it would be 0, hence it would not move, but would remain there forever unless disturbed).

According to Newton's second law of motion, force is the action that causes
a change in the motion of a particle or sets it in motion from rest.
Consider a particle of mass moving in one dimension so that its
position is . If a force acts on the particle, it will be set
in motion, so that is no longer a fixed number but a function
of time . At any instant in time, the particle will have
a velocity defined to be the rate of change of with respect to
time

In general is not fixed but is also a function of time . If is not constant, in fact, then the particle also has an acceleration defined as the rate of change of with respect to time

Since , it follows that

Newton's second law of motion states that the force acting on the particle is directly proportion to this acceleration, with the mass as the constant of proportionality

Since motion generally occurs in three rather than one dimension,
position, velocity, acceleration, and force are all vector quantities.
The position is usually denoted , and has three
components
. Similarly velocity has
components
, accleration
,
and
. In vector notation, Newton's second law reads

Hence, if is not a fixed vector but a function of time, this means that each component of is a function of time .

**Work**: Consider again a particle of mass moving in one spatial
dimension. If a force is needed to move the particle from position
to position , then *mechanical work* has been done on the
particle. Since, as we have seen, can be a function of ,
the general definition of work is the area under the force function
between and , i.e. the integral

However, since , this becomes

Now if the velocity of the particle at is and at , it is , then by energy conservation

Rearranging this gives

Hence, the work is also equal to the change in kinetic energy

a result known as the