## 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 Fw 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 energyRemember 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 (Fw) 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 (Fw) 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 = Fwh

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/s2 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 mv2
Work = KE

Fd = ½ mv2

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?

If you need help on ratio problems click the link below:

Rule of Ones: analyzing equations to determine how other variables change

### 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.

MEi = MEf

### mghi + ½ mvi2 = mghf + ½ mvf2

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.

MEi = MEf + 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?

### Mechanical Energy Practice Quiz

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Mechanical Energy Quiz

Do you know your mechanical energy? Take our quiz to find out

1 / 14

What happens to kinetic energy as an object falls under ideal conditions?

2 / 14

What happens to potential energy as an object falls under ideal conditions?

3 / 14

What happens to mechanical energy as an object falls under ideal conditions?

4 / 14

Potential energy is dependent on __________________.

5 / 14

Kinetic energy is dependent on __________________.

GPE = mgh or GPE = Fwh

You are given weight not mass so use this form:

GPE = mgh = (50)(3.5) = 175 J

6 / 14

How much potential energy is in a  50kg boulder 3.5m off the ground?

7 / 14

How much potential energy does a 568 N ballerina have when she leaped 0.35 meters off the ground?

8 / 14

How much more potential energy do you have when an object is five times higher?

9 / 14

Mechanical Energy is composed of ____________.

10 / 14

When energy is lost from the system its lost as _______________.

11 / 14

How much kinetic energy does a 30 kg boar have when running 6 m/s?

12 / 14

How much does kinetic energy change when the velocity of an object doubles?

13 / 14

If a car has a mass of 750 kg, how much force is required to stop the car if it was traveling 12.5 m/s and took 10 m to stop?

14 / 14

A pool ball is flung off of a 0.68 m high table and the ball hits the floor with a speed of 6.0 m/s.  How fast was the ball moving when it left the pool table?