# Kinetic Energy and Potential Energy

### Written by tutor German C.

The Law of Conservation of Energy states that energy cannot be created or destroyed. In other words, the total energy of a system remains constant. This is an important concept to remember when dealing with energy problems. The two basic forms of energy that we will focus on are kinetic energy and potential energy.

## Kinetic Energy

**Kinetic energy** is energy that comes from motion. In other words, objects that are moving have something that is referred to as kinetic energy. Since kinetic energy is based on motion, it is always a positive value. If it is not in motion, the kinetic energy of that object is zero. Kinetic energy can never be a negative value. Kinetic energy can be quantified as one half of the mass times the velocity squared (KE = ^{1}/_{2}*m*v²). In SI units, the mass should be in kilograms (kg), and the velocity in meters per second (m/s). In English units, the mass should be either pound mass (lbm) or slug, and the velocity in feet per second (ft/sec).

## Potential Energy

Potential Energy is just like it sounds, it is energy that is related to an object's potential. Potential Energy can be quantified as mass times gravity times height (PE=m*g*h). The unit for mass should be either kilogram or pound mass depending on the unit system. Gravity is a constant, 9.81 ^{m}/_{s²} in SI units or 32.2 ft/sec² in English units. Gravity is an acceleration, it can be described as meters or feet per second per second or the change in velocity per second. Lastly, the height, has units of meters in SI units and feet in English units. It is important to note that height is considered a relative quantity. In other words, when looking at potential energy, the first step is to establish a datum or origin. Meaning, an elevation where the height is zero needs to be defined. For example, the floor can be defined as a height of zero. However, the zero point (datum) does not have to be the floor, it can be any point, but it must not be changed once it is defined. It is also important to note that potential energy can be positive, zero or negative. For example, if the datum is defined as the top of a table, and the object is on the floor, that object has a negative potential energy since the height below the top of the table.

Now that the kinetic energy and potential energy have been defined, we can now apply the Law of Conservation of Energy. In other words, the kinetic energy plus the potential energy equals a constant (KE+PE=Constant).

Let's imagine a simple energy problem. There is an object that travels from one point to another. We will call the first point the starting point (1), and the second point the ending point (2). Without considering anything else we can set up the basic equation as KE_{1}+PE_{1}=KE_{2}+PE_{2}. Let's simplify this even more; let's say the object starts at rest (KE_{1}=0) and let's say we define the datum at the ending point (PE_{2}=0). Now we can say that PE_{1}=KE_{2}. The energy has literally been converted from potential energy to kinetic energy. Note that the total energy at point 1 is equal to the energy at point 2, the energy has changed forms, but was not created or destroyed.

It is also good to know that energy is considered path independent. In other words, it does not matter what path the object took to get from point 1 to point 2, the energy at point 1 and point 2 did not change. The only thing that matters is the difference in height of the two points; as long as the height does not change from scenario to scenario, the solution does not change.

## Kinetic and Potential Energy Examples

Imagine a ball rolling across the floor from one point to another.

Let's say the ball has a mass of 3 kg and is traveling at 2 ^{m}/_{s}

At point 1:

the kinetic energy = ^{1}/_{2}**m**v_{1}^{2} = ^{1}/_{2}(3kg)(2^{m}/_{s})^{2}

=6 kg*^{m2}/_{s2} = 6 N*m = 6 J

where **m** = mass in kilograms

v = velocity in ^{m}/_{s}

m = unit of length, meter

s = unit of time, second

N = unit of force, Newton

1 N = 1 kg*^{m}/_{s2}

J = unit of energy, Joule

1 J = 1 N*m

Also, the potential energy = **m**gh_{1} = 0, since h = 0

g = acceleration due to gravity = 9.8^{m}/_{s2}

h_{1} = height relative to datum (origin), in meters

At point 1 the total energy = KE_{1} + PE_{1} = 6 + 0

= 6 Joules

Note at point 2:

PE_{2} also = 0, therefore KE_{2} = 6

PE_{2} = 0 because h_{2} = 0 since floor is level.

If KE_{2} = KE_{1} and mass does not change,

v_{2} = v_{1} = 2^{m}/_{s}

### Another example

Now let's say h_{1}≠h_{2}, let's say h_{2} = -2m

h_{2} = -2m means that point 2 is 2 meters below point 1.

A, B, and C represent three possible paths. Note: the path taken has no influence on the solution.

Recall that KE = 6 J, PE = 0 J

Since total energy remains constant

KE_{1} + PE_{1} = KE_{2} + PE_{2}

6 + 0 = KE_{2} + PE_{2}

We can find PE_{2} since h_{2} = -2m

PE_{2} = **m**gh_{2} = (3kg)(9.8^{m}/_{s})(-2m) = -58.8J

Now 6 + 0 = KE_{2} - 58.8

Now solve for KE_{2}

6 + 58.8 = KE_{2}

KE_{2} = 64.8 J

To find v_{2}

KE_{2} = ^{1}/_{2}**m**v_{2}^{2} = ^{1}/_{2}(3kg)(v_{2})^{2} = 64.8

v_{2}^{2} = ^{2}/_{3}(64.8) = 43.2

v_{2} = 6.57 ^{m}/_{s}

An increase of 4.57 ^{m}/_{s}

## Other Types of Energy

Energy is not only limited to kinetic and potential energy. There are many different forms of energy. However, in basic Physics, the other forms are ignored in order to simplify the subject of energy equations. The other forms of energy are usually introduced at the College level.

For example, work introduced to a system can and affect the total energy of that system. For example, imagine a piece of furniture in your back yard. It is sitting there and not moving. Since it is not moving, it does not kinetic energy. Also, since it is on the ground, it cannot go any lower. In this case, both the potential and kinetic energy are zero. Now what happens if someone comes and picks up the couch. The potential energy of that couch has changed, but it did not come from the kinetic energy. So where does that energy come from? Was it created? Did we violate the Law of Conservation of Energy? The answer is no, we did not violate any laws of Physics. Originally, the couch is considered as an isolated system in our example. The person is an external entity that introduced energy into that system by picking up the couch. In Physics terms, the person did work on the system. Work is considered as a form of energy. Work can be quantified a force times a distance.

However, it is not that simple. The force and distance must be pointing in the same direction. For example, a weight is a force. For example, work can be done by gravity. Actually work done by gravity is another way to define potential energy. Weight is mass times gravity. Gravity always points down, therefore the force points down as well. For gravity to do work, an object must either rise or fall. In this case, no work is being done if an object moves side to side. This concept is better understood by understanding vectors.

To help clarify a vector, we will compare velocity to speed. Speed is a magnitude, for example 60 miles per hour. Note that speed does not define a direction. Velocity is a vector, meaning it has a magnitude and direction. For example, a velocity can be 60 miles per hour in the y-direction. Vectors are best defined in projectile problems also referred to as kinematics equations.

Another case where energy might not remain constant is in a collision problem. Let's say a person holds an apple at shoulder height. At this point, the apple has potential energy but no kinetic energy. Then the person releases the apple allowing it to begin falling. As it is falling, the energy is being converted from potential energy to kinetic energy. Just before it strikes the ground, almost all of the potential energy has been converted to kinetic energy. But what happens when it actually hits the ground? When the apple hit the ground, it is actually colliding with the ground, and that kinetic energy gets absorbed by the ground during the collision.

During a collision, momentum is conserved. Depending on the coefficient of restitution, the two objects involved in a collision actually distribute the moment before and after the collision based on their material properties. For example, imagine a person is holding two objects in their hands. In one hand, the person is holding a basketball. In the other hand, a person is holding a bag filled with sand. If the person drops both objects onto a concrete floor, the basketball will bounce, but the bag of sand will just plop on the ground and lay there. During the collision, the basketball retains most of its momentum while the bag of sand gives it all up. There is a reason why this happens; the coefficient of restitution between the concrete and the basketball is different from the coefficient of restitution between the concrete and the bag of sand.

## An Example Using Work

Now, let's look at the original example again, except a force will be applied from point 1 to point 2, like someone tied a string to the ball.

Let's say the distance from point 1 to point 2 is 20 meters and the force is 7 Newtons.

*Note the angle between the force and path of travel is 35°.

Also recall KE_{1} = 6 J, PE_{1} = 0

Again, since flat, PE_{1} = PE_{2} = 0

Very important: total energy is not constant because of the applied force. This applied force introduces a new energy through work.

where

w = work, in Joules

F = applied force, in Newtons

d = distance from point 1 to point 2, in meters

· = dot product

-> = defines values as vectors

Therefore:

where Θ = angle between force and path.

Finally W_{1-2} = [7(cos(35°))N](20m) = 114.68J

The new equation

KE_{1} + PE_{1} + W_{1-2} = KE_{2} + PE_{2}

6 + 0 + 114.68 = KE_{2} + 0

120.68 J = KE_{2}

Lastly, to find v_{2}

KE_{2} = ^{1}/_{2}**m**v_{2}^{2}

120.68 = ^{1}/_{2}(3kg)(v_{2})^{2}

80.45 = v_{2}^{2}

v_{2} = 8.97 ^{m}/_{s}

An increase of 6.97 ^{m}/_{s} due to applied force across 20 m from point 1 to point 2.