# What Is The Energy Equation (Formula)

In physics, there are two main kinds of mechanical energy: *potential energy *and *kinetic *energy. Both potential and kinetic energy are important concepts for understanding the motion of objects and the physical effects that they can produce. The two equations that describe the potential energy (PE) and kinetic energy (KE) of an object are:

**PE** = *mgh*

**KE** = ½*mv²*

where *m *is the mass of the object, *g* is the height of the object, *g *is the gravitational field strength (9.8m/s²), and *v *is the average velocity of the object. The formula for potential energy states that the potential energy stored in an object is directly proportional to the position of an object with respect to a defined 0 point. The formula for kinetic energy states that the kinetic energy of a body is directly proportional to the velocity of a body.

There is also a special equation for *elastic potential energy*, which describes the energy stored in a compressed or stretched elastic material, like a spring, trampoline, or a bow with a nocked arrow. The elastic potential energy equation is:

**PE _{spring}** =

*kx*

where *x *is the magnitude of compression or stretching and *k *is a constant of proportionality called the *spring constant*. The value of *k *is dependent on the particular physical properties of a given spring. This mathematical expression is sometimes referred to as *Hooke’s Law*, named after the English scientists Robert Hooke who first formulated the principle in 1660. Essentially, Hooke’s law states that the magnitude of deformation of an elastic material is directly proportional to the force impressed on that material. All three kinds of mechanical energy potential, elastic potential and kinetic are expressed in the same unit, the **joule (J)**.

## Potential And Kinetic Energy: The Basics

Energy comes in many different forms, electromagnetic, chemical, solar, heat, nuclear and mechanical to name a few. Despite all these different manifestations, at their core, all kinds of energy are the same thing: energy is simply the capacity of a system to perform work. All the different forms of energy can be converted into one another like how a combustion engine converts heat energy into mechanical energy, or a nuclear reactor converts nuclear energy into electrical energy.

Mechanical energy is understood as the energy associated with the movement and position of physical objects; i.e. mechanical energy refers to the capacity of an object to produce work in virtue of its position and movement. Potential and kinetic energy are two kinds of mechanical energy.

### Potential Energy (PE)

Potential energy could be thought of as the energy that is stored in a body when it is moved to a certain position. If one takes a pendulum, raises the bob in the air, and holds it there, the bob now has a potential energy that is proportional to the height it was raised from its resting point. Similarly, if one stretches a spring so that it is taut and holds it there, the spring has potential energy that is stored in the stretched coils of the spring. Potential energy is just that; a *potential *to do work. When I stretch the spring and hold it, the spring is not doing any work. However, if I release the spring, it will rapidly contract due to the potential energy stored in the stretched spring. With the pendulum bob, the potential energy of the bob refers to its disposition to fall under the force of gravity back to its resting spot.

Say I raise a 30 kg block up to a height of 8 meters above the ground. Using our equation, we can determine the total potential energy possessed by the block:

PE = *mgh*

PE = 30kg • (9.8m/s²) • 8m = **2352 J**

Thus, at 8 meters above the ground, a 30kg block has a total potential energy of **2352 J**

### Kinetic Energy (KE)

Kinetic energy is understood as the energy an object possesses when it is moving. Intuitively, one can think of kinetic energy as the energy possessed by a bowling ball in virtue of its motion down the lane, or the energy possessed by a baseball that is launched from a pitcher’s arm. If potential energy is a *potential* the kinetic energy is *actual*. An object is able to change the position or movement of another object in virtue of its kinetic energy, as demonstrated when a moving object collides with another and changes the other object’s motion. The kinetic energy of an isolated system that undergoes elastic collisions is always conserved, meaning that the total kinetic energy remains constant.

As an example to illustrate kinetic energy, let’s say that a 17 kg sphere is moving in a straight line with a velocity of 5 m/s. Using our equation, we can determine the kinetic energy of the sphere:

**KE** = ½*mv²*

KE = ½ • 17kg • (5m/s)² = **212.5 J**

So the sphere has a total kinetic energy of **212.5 J**

All kinds of energy can be converted into one another. The same holds true for potential and kinetic energy. Say I place a book on top of a table so that it is at rest. In this position, the book has gravitational potential energy that is proportional to the height it is off of the ground. When the table is removed so that the book can fall to the ground freely, the potential energy stored in the book is converted into kinetic energy as the book begins to move. In the same vein, when I stretch a spring, the potential energy is stored in the stretched coils of the spring. When I release the spring, that potential energy is converted into kinetic energy as the spring snaps back to an equilibrium state.

## Relationship Between PE And KE

In any physical system, the total mechanical energy of that system is equal to the sum of the potential and kinetic energies of that system. The total energy of a mechanical system is always conserved under elastic collisions. Because the total energy of a system is conserved, if there is a change in the potential energy of a system, there must also be a corresponding change in the kinetic energy of a system that will keep the total mechanical energy constant. For instance, if the potential energy of a system decreases by 20J, then the kinetic energy of that system must increase by 20J to keep the total energy constant. Knowing this, we can derive a mathematical relationship:

ΔE = ΔKE + ΔPE

Since total energy is always conserved, we can set ΔE = 0 so,

0 = ΔKE + ΔPE

**ΔKE = −ΔPE **

That is, the change in the magnitude of kinetic energy is equal to the opposite of the change in the magnitude of potential energy.

Using this mathematical relationship, we can figure out how to convert potential energy into kinetic energy and how to figure out other properties of moving objects. Let’s consider a simple example:

A person in a helicopter drops a 90 kg package from a height of 180 m.

- (a) what is the potential energy of the package before it is released?
- (b) what is the kinetic energy of the package just before it hits the ground?
- (c) what is the velocity of the package just before it hits the ground?

For (a) since the package is held at rest 180 meters above the ground the total potential energy of the package in that position is:

PE = 90kg(9.8m/s²)(180m) = **158760 J**

So the answer to (a) is, at its highest point, the package has a total kinetic energy of **158760 J.**

For (b), we know that once the package has hit the ground, it will have converted all of its potential energy into kinetic energy. We know that the original potential energy of the package/helicopter system is 158760 J. Because the system is not moving at its initial point, the kinetic energy is 0. After the package has fallen, it will convert all of that energy into kinetic energy. Thus, the change in potential energy after the package has fallen is ΔPE = −158760 J. Using our formula to relate the change in PE to the change in KE we get:

**ΔKE = −ΔPE **

ΔKE = −(−158760)

ΔKE = **158760 J**

Because the package has converted all of its potential energy during the fall, at the end of the fall its kinetic energy will be equal to the potential energy before the fall. So for (b), just before it hits the ground, the package has a kinetic energy of **158760 J.**

We can figure out the answer to (c) by using the formula that relates kinetic energy to velocity. If just before the package hits the ground it has a kinetic energy of 158760 J, then its velocity at that point can be determined:

**KE** = ½*mv²*

158760 = ½(90kg)(v²)

3528 = *v*²

**v = 59.40 m/s**

Using our equation that relates kinetic energy to velocity, we can determine that just before the package hits the ground, it is traveling with a velocity **59.40 m/s**.

In summation, mechanical energy refers to the energy possessed by an object in virtue of its position and motion. Potential energy is defined as the energy possessed by an object in virtue of its position, and kinetic energy is the energy of an object in virtue of its motion. the two formulas for PE and KE are:

**PE** = *mgh*

**KE** = ½*mv²*

PE and KE can further be related as the sum of PE and KE in a system is equal to the total mechanical energy in that system. Because energy is always conserved, a change in potential energy must be balanced out by a corresponding change in kinetic energy, so:

**ΔKE = −ΔPE**

Using these mathematical relationships, one can solve a whole host of problems related to the position, motion, and energy of a system.