# Energy

There’s no precise definition, but an approximate definition of energy is “the ability or the capacity to do work”.

- property of matter and waves that can be transferred among objects and can change its form

The joule (J) is the SI unit for energy.

Why do we need to understand energy?

Some problems require the study of energy, like in rocket propulsion, how much fuel we need to burn. Newton’s law of motion cannot answer this question.

Surprisingly, energy was only fully understood at the beginning of the 19th century, 150 years after Newton’s

Principia Mathematicapublication.

### Work

The discussion of energy comes first by discussing and understanding work.

### Kinetic Energy

$K=21 mv_{2}(kinetic energy)$ This formula is actually derived from $v_{f}=v_{i}+at$ → $v_{f}−v_{i}=2ax$ combined with the definition of work.

We get the *Work-Energy Theorem*, which states that the work done on an object is equal to the change in the object’s kinetic energy.
$W_{F}=21 mv_{f}−21 mv_{i}=ΔK$

### Potential Energy

The *potential energy* of an object is its capacity/ability to do work. Denoted by the letter $U$.

$ΔU=−∫_{a}F(r)⋅dr$

**Gravitational Potential Energy**
$U_{g}=mgh$
$ΔU_{g}=mgΔh$
**Elastic Potential Energy**
$ΔU_{spring}=21 kx_{f}−21 kx_{i}$
$U_{spring}=21 kx_{2}$

### Conservation of Mechanical Energy

See also - Law of Conservation of Energy $ΔK=W_{net}=W_{nc}+W_{c}$ where $nc$ stands for non-conservative forces, and $c$ stands for Conservative Force.

$ΔK+ΔU=W_{nc}$

In the absence of non-conservative forces, $ΔK+ΔU=0$

Not to be confused with Cal (also known as kcal).