## Mechanical Energy

**Mechanical energy** is energy in motion (kinetic energy) or energy that can become motion (potential energy).

**Total mechanical energy is the kinetic energy plus potential energy**. Learn the equations for kinetic and potential energy and see various ways to use them.

Notice the following variables in equations on this page. It is important to be familiar with the variable symbol and unit when working through problems.

### Mechanical Energy Variables

Name |
Variable |
Unit |
Unit Abbreviation |

Mechanical Energy |
ME | Joule | J |

Potential Energy |
PE | Joule | J |

Kinetic Energy |
KE | Joule | J |

Mass |
m | Kilogram | kg |

Acceleration Due To Gravity |
g | Meters per second squared | m/s2 |

Velocity |
v | Meters per Second | m/s |

Height or distance |
h or d | Meters | m |

Weight |
F_{w} |
Newtons | N |

**Forms of Energy**

Different forms of energy we have in nature include:

- Chemical Potential Energy
- Mechanical (ME): Kinetic Energy (KE) & Potential Energy (PE)
- Light
- Heat
- Nuclear
- Electric

In this unit we will concentrate on **mechanical energy**. If a ball falls from the top of a cliff it starts with potential energy due to its height. As gravity does work on the ball it falls to the ground converting potential energy to kinetic energy. Joules of energy lost from potential will be gained by kinetic energy.

**Doing Work to Store Potential Energy**

**You do work to create potential energy**. **Remember work** equals a force applied over a distance (**W=Fd**).

Gravitational potential energy (**GPE**) is a very common type of potential energy in problems. Often problems just call it potential energy. GPE can be stored by lifting an objects weight (F_{w}) a certain distance up or height. Therefore you can calculate GPE of an object the same way seen in the image.

### Example Problems:

**1. How much gravitational potential energy do you have when you lift a 15 N object 10 meters off the ground?**

**2. How much gravitational potential energy is in a 20 kg mass when 0.6 meters above the ground?**

**Gravitational Potential Energy**

Because weight (F_{w}) is equal to mass times gravity either of the equations below can be used to solve for GPE. Be careful to make a givens list and pick the right equation.

**GPE = F _{w}h**

**GPE = mgh**

Note: When picking the formula **units help determine which form of GPE to use**. The unit for weight is a newton (N), mass is in kilograms (kg), height is meters (m). The acceleration due to gravity (g) is 9.8 m/s^{2 }most places on earth.

**Work Done to Stop Motion**

Work is also done to stop and object from moving and would be equal to kinetic energy (KE) we will see later in this page.

Work = Fd and KE = ½ mv^{2}

Work = KE

**Fd = ½ mv ^{2}**

**Example Problems**

**3. How much gravitational potential energy does a 35 kg boulder have when 30 meters off the ground?**

**4. How many times greater is an objects potential energy when three times higher?**

**Kinetic Energy**

**Kinetic energy is energy in motion**. The **velocity** of an object is very important in determining its kinetic energy. The formula for kinetic energy is as follows:

Kinetic energy and mass Is directly related. Kinetic energy and velocity is also but has an exponential effect because it’s squared.

### Example Problems

**5. How much kinetic energy does a 0.15 kg ball thrown at 24 m/s have?**

**6. How many times greater is the kinetic energy of a ball that is going five times faster?**

### Potential Energy Becoming Kinetic

Potential energy becomes kinetic when an object falls from rest and lands directly on the ground. The mechanical energy does not change.

Here are forms of the conservation of energy formula you would use in this situation. The actual one depends on the information given and what is asked. Making a givens and unknown list is very important.

#### All Initial Potential Energy Becoming All Kinetic

Be aware that problems asking for final velocity are while an object is moving before hitting the ground.

The **Conservation of Energy Law** states that you cannot create or destroy energy. Therefore, it can only be converted to heat and lost from the system but not destroyed. A **system** is any unit you are studying like a roller coaster.

**Mechanical energy at the beginning equals the ending mechanical energy **in ideal situations where you ignore heat.

You can derive additional equation forms looking at the picture above. This is because mechanical energy is made of potential and kinetic and each of those have their own equation.

### Conservation of Energy Equation Forms in Ideal Situations (Most Questions)

**ME _{i} = ME_{f}**

**PE**_{i} + KE_{i} = PE_{f} + KE_{f}

_{i}+ KE

_{i}= PE

_{f}+ KE

_{f}

**mgh**_{i} + ½ mv_{i}^{2} = mgh_{f} + ½ mv_{f}^{2}

_{i}+ ½ mv

_{i}

^{2}= mgh

_{f}+ ½ mv

_{f}

^{2}

**In reality energy lost will be in the form of heat**. The conservation of energy formula can be arranged many ways with a few common ones seen below.

Use this equation if a problem asks for how much heat is lost. ME can be broken into PE + KE as well.

**ME _{i} = ME_{f} + Heat**

**Setting Up Conservation of Energy Problems**

Once you realize a problem is talking about two time periods (before and after situation) take an additional step. Set up a initial and final givens list as seen to the right.

**Example:** A 85kg roller coaster cart **is traveling** 4 m/s at the top of a hill 50 meters off the ground. How fast is it traveling **at top of a second hill** 20 meters off the ground?

### Mass Cancels Out of "Some" Conservation of Energy Problem Situations

- When given velocity or height and have the other before and given velocity and height after mass can cancel out
- When given a velocity and height before and asked for velocity or height after, and given the other after, mass can cancel out.
- You can still use the mass if a given in a problem but don't need it. Mathematically it will cancel itself out.

### Example Problems

**7. How much kinetic energy does a 1.2 kg ball have the moment it hits the ground 3.5 meters below when it starts from rest?**

**8. How fast is a 1.2 kg ball traveling the moment it hits the ground 3.5 meters below when it starts from rest?**

**9. A 3.5 kg ball falls from a height of 12 meters. How fast is it traveling when its still 5 meters off the ground?**

**10. ****An 85 kg roller coaster cart is traveling 4 m/s at the top of a hill 50 meters off the ground. How fast is it traveling at top of a second hill 20 meters off the ground?**

**(Click Here for All the Mechanical Energy Example Problems and Solutions)**

### Mechanical Energy Practice Quiz

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