When an object such as a ball or a car sits at the top of a hill, it has potential energy. Potential energy is the stored energy an object has because of its position or shape. In the case of an object on top of a hill, the object has potential energy due to gravity pulling it down the hill. As the object starts rolling or sliding down the hill, this potential energy gets converted to kinetic energy, which is the energy of motion. The faster the object moves down the hill, the more kinetic energy it gains.

## Definitions of Potential and Kinetic Energy

Potential energy is energy that is stored in an object due to its position or shape. For example, a ball sitting on a table has potential energy since gravity could make it fall off the table. Other examples are a compressed spring, stretched rubber band, or water held behind a dam. The key is that potential energy describes stored energy due to an object’s position.

Kinetic energy is energy that an object possesses due to being in motion. The faster an object moves, the more kinetic energy it has. Some examples are a moving car, flowing river, or baseball in flight. Kinetic energy depends on both the mass and speed of the moving object.

### Equation for Potential Energy

The potential energy of an object near Earth’s surface can be calculated using the equation:

PE = mgh

Where:

PE = Potential Energy (in Joules)

m = mass of the object (in kilograms)

g = acceleration due to gravity (9.8 m/s2)

h = height of the object (in meters)

So an object with more mass at a greater height above the ground will have more potential energy. This energy is just waiting to be converted to kinetic energy if the object starts moving down.

### Equation for Kinetic Energy

The kinetic energy of a moving object can be calculated using:

KE = 1/2 mv2

Where:

KE = Kinetic Energy (in Joules)

m = mass of the object (in kilograms)

v = velocity or speed of the object (in m/s)

So an object with more mass moving at a higher velocity has more kinetic energy. This energy is directly related to the object’s motion.

## Potential to Kinetic Energy Conversion

When the object at the top of the hill starts rolling down, its potential energy gets converted into kinetic energy. As it rolls faster down the hill it gains more kinetic energy. The total energy remains constant, being transferred between potential and kinetic.

Some key points:

- At the top of the hill the object has maximum potential energy and zero kinetic energy
- As it rolls down it loses potential energy but gains kinetic energy
- At the bottom of the hill the potential energy is at a minimum and kinetic energy is maximized
- As the object rolls up the next hill the kinetic energy converts back into potential energy
- The total energy is conserved, just transferring between potential and kinetic

This transfer of energy explains how roller coasters, pendulums, and springs work. The cycling back and forth between potential and kinetic energy allows for oscillating motions.

### Table of Potential vs Kinetic Energy

Location | Potential Energy | Kinetic Energy |
---|---|---|

Top of first hill | High | Zero |

Rolling down | Decreasing | Increasing |

Bottom of first hill | Minimum | Maximum |

Rolling up next hill | Increasing | Decreasing |

This table summarizes the exchange between potential and kinetic energy.

## Factors Affecting Potential Energy

There are two main factors that determine an object’s potential energy:

### 1. Mass

Heavier objects have more potential energy since gravity acts on them with more force. Consider two balls, one twice as heavy as the other, at the same height on a hill. The heavier ball has twice the potential energy since gravity pulls on it harder. This means it can gain more kinetic energy rolling down the hill.

### 2. Height

Greater height above the ground means more potential energy. Going back to the balls, if one is twice as high on the hill, it has twice the potential energy. This is because it can fall twice the distance, accelerating more and gaining more kinetic energy.

Potential energy increases exponentially with height. Small differences in height make a big difference in potential energy. This is why heights are very important when calculating potential energy.

## Factors Affecting Kinetic Energy

The kinetic energy of a moving object also depends on two primary factors:

### 1. Mass

Just like with potential energy, heavier objects have greater kinetic energy at the same speed. This makes sense since it takes more effort to get a heavier object moving. Two cars moving at the same speed don’t have the same kinetic energy if one car is much heavier.

### 2. Speed

Faster moving objects have much greater kinetic energy. Kinetic energy actually goes up with the square of velocity. So doubling speed does not double kinetic energy, it actually quadruples it. This is why small increases in speed result in large increases in kinetic energy.

## Examples and Applications

### Roller Coaster

Roller coasters provide an excellent real-world demonstration of potential and kinetic energy conversions. At the top of the first hill, the roller coaster train has maximum potential energy. As it rolls down, this converts into kinetic energy, reaching top speed at the bottom. It then rolls up the next hill, losing kinetic energy and regaining potential energy. The cycles of hills and drops allow the roller coaster to seem to defy gravity without any outside energy needed.

### Pendulum

A swinging pendulum also nicely illustrates the conversions between potential and kinetic energy. As the pendulum swings back and forth, it is constantly exchanging energy between potential at the extremes of the swing, and kinetic energy as it moves fastest at the bottom. The pendulum will eventually come to rest due to friction dissipating energy.

### Hydroelectric Dam

Dams provide a great real-world system to analyze potential and kinetic energy. The water held behind the dam has potential energy that can be released and converted into electricity. As the water falls down through the turbines, it loses potential energy but gains kinetic energy. This kinetic energy is used to turn the turbines and generate electricity. Dams are an excellent illustration of potential energy being captured and turned into useful work.

### Bouncing Ball

The bouncing of a ball demonstrates potential and kinetic energy transfers. At the top of the bounce, it has maximum potential energy. As it falls, this converts to kinetic energy until it hits the ground. Some energy is lost, but much of the kinetic energy turns back into potential energy as the ball is compressed on impact, allowing it to rebound upwards. This cycle repeats until the energy is dissipated through friction and the ball comes to rest.

## Practice Problems

Here are some practice problems to work through involving potential and kinetic energy:

### Problem 1

A 5 kg ball sits at the top of a 3 m high hill. What is its potential energy?

Using the potential energy equation:

PE = mgh

m = 5 kg

g = 9.8 m/s2

h = 3 m

PE = (5 kg)(9.8 m/s2)(3 m)

PE = 147 J

### Problem 2

A 10,000 kg truck is moving at 20 m/s. What is its kinetic energy?

Using the kinetic energy equation:

KE = 1/2 mv2

m = 10,000 kg

v = 20 m/s

KE = 0.5(10,000 kg)(20 m/s)2

KE = 2,000,000 J

### Problem 3

A ball falls off a shelf that is 2 m high. Right before hitting the ground its speed is 5 m/s. What fraction of the initial potential energy has been converted to kinetic energy?

Initial potential energy:

PEi = mgh

PEi = (1 kg)(9.8 m/s2)(2 m)

PEi = 19.6 J

Final kinetic energy:

KEf = 1/2 mv2

KEf = 0.5(1 kg)(5 m/s)2

KEf = 12.5 J

Fraction of energy converted:

KEf/PEi = 12.5 J / 19.6 J = 0.64

So 64% of the potential energy was converted to kinetic energy. The rest was lost to friction, sound, deformation, etc.

## Conclusion

In summary, when an object sits at the top of a hill it has potential energy due to gravity. As it rolls down, this potential energy gets converted into kinetic energy, the energy of motion. The factors that affect potential energy are mass and height, while mass and speed determine kinetic energy. Real-world systems like roller coasters, pendulums, dams, and bouncing balls all illustrate the cycling between potential and kinetic energy through conservation of energy. Understanding these concepts allows analyzing and designing systems that take advantage of potential and kinetic energy transfers.