### Forms of Energy

There are 3 main types of energy.

- Kinetic (any mass that has velocity)
- Potential (any mass has stored energy)
- Work (all other types of energy that isn’t potential or kinetic).

Note that *Elastic *Energy also called spring potential energy is a type of potential energy.

Note that *Mechanical *Energy is only the sum of Kinetic and Potential energy.

Now lets dive into each type of energy and cover some misconceptions.

#### Kinetic Energy

Kinetic energy (KE) is the energy of motion, and is given by the formula:

[katex] KE = \frac{1}{2}mv_2 [/katex]

For example, a car moving at a high speed has more kinetic energy than a car moving at a slow speed.

Now to speed something up would mean to increase kinetic energy, which would come from an external source (Work energy). This is why [katex] \Delta KE = \text{Work} [/katex]

#### Potential Energy

Potential energy is the energy of position or configuration. It is stored energy that an object possesses due to its position or arrangement. There are several types of potential energy, such as gravitational potential energy, elastic potential energy, and chemical potential energy.

**Gravitational potential energy** **(PE _{g})** is the energy of an object due to its height. It is given by the formula

[katex] PE_g = mgh [/katex]

Where “m” is mass, “g” is the value of gravity, and “h” is the height of the object.

For example, a rock at the top of a hill has more gravitational potential energy than a rock at the bottom of the hill.

**Elastic (or Spring) potential energy** is the energy stored in a stretched or compressed spring. It is given by the formula

[katex] El = \frac{1}{2}kx^2 [/katex]

Where “k” is the spring constant (how stiff the spring is), and “x” is the amount of stretch/compression in meters.

For example, a spring that is stretched more has more elastic potential energy than a spring that is stretched less.

**Chemical potential energy** is the energy stored in the chemical bonds of a substance. It is the energy that is released or absorbed during a chemical reaction. For example, gasoline has a high chemical potential energy, which is released when it is burned to produce energy.

#### Work Energy

[katex] \text{Work} = Fd [/katex]

In simple terms work is the force applied over a certain distance. Pushing a box is a simple example of positive work being done. The force of friction is an example of negative work being done.

Keep in mind the other definition of work:

[katex] \text{Work} = \Delta KE [/katex]

There are two important things to note. (1) Energy is a scalar. Positive and negative simply mean that the object is either gaining (+) or loosing (-) energy. (2) Force and displacement MUST be parallel to each other in order for there to be work energy.

#### Other Types of Energy

Here are a few more types of energy, not covered in the AP Physics 1 exam.

**Thermal energy** is the energy of heat. A common example is friction. It is the energy that is transferred from one body to another as a result of a temperature difference. Thermal energy is related to the temperature of an object, with hotter objects having more thermal energy than cooler objects.

**Electrical energy** is the energy of electric charge. Energy is transferred through an electric circuit due to the force applied over a distance (Work done) on charged particles.

### Energy Conservation

The law of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. This means that the total amount of energy in a closed system remains constant, even if the energy changes form.

No change in energy → [katex] \Delta E = 0 \rightarrow E_f – E_i = 0 \rightarrow E_f = E_i [/katex]. In simple terms, this means the amount of energy a system starts with should be the exact same amount it ends with.

For example, consider a ball being thrown into the air. The ball has kinetic energy due to its motion, and it also has gravitational potential energy due to its height. As the ball rises, its kinetic energy decreases and its gravitational potential energy increases, but the total amount of energy remains constant.

### Framework for Solving Energy Problems

When solving energy problems, it is important to identify the forms of energy involved and use the appropriate formulas to calculate the energy.

- Start with the Law of Conservation of Energy which states → [katex] E_i = E_f [/katex]
- Read the scenario and identify the initial forms of energy. Write it down under [katex] E_i [/katex]
- Identify the final forms of energy. Write it down under [katex] E_f [/katex]
- Substitute the equations for each form of energy. Rearrange your equation and solve for asked variable.

Let’s practice this framework with 3 commonly asked AP Physics 1 questions on Energy

#### Example Problem #1

A ball is thrown straight up into the air with an initial velocity of 10 m/s. What is the maximum height reached by the ball?

FRAMEWORK | Solving the Problem |

Step #1. Law of Conservation of energy | [katex] E_i = E_f [/katex] |

Step #2. Identification of initial form of energy | [katex] E_i = KE [/katex] (since the ball is moving as it’s initially thrown up) |

Step #3. identification of final form of energy | [katex] E_f = PE [/katex] (at the maximum height all the kinetic energy transforms into gravitational potential energy |

Step #4. Substitute KE and PE into [katex] E_i = E_f [/katex] | [katex] E_f = E_i \rightarrow KE = PE \rightarrow \frac{1}{2}mv^2 = mgh [/katex] |

Step #5. Rearrange and solve of unknown | Solve for height → [katex] h = \frac{v^2}{2g} [/katex] |

## Show Further Explanation

The maximum height of the ball is determined by the balance between its initial kinetic energy and the gravitational potential energy it gains as it rises. As the ball rises, its kinetic energy decreases and its gravitational potential energy increases, until they are equal at the maximum height. At this point, the ball stops rising and begins to fall back down.

#### Example Problem #2

A 0.5 kg block is pushed up a frictionless ramp with a constant force of 5 N. The block starts at rest at the bottom of the ramp and reaches a height of 2 meters at the top. How much work is done on the block by the applied force?

Famework Ste | Solving the Problem |

Step #1. Law of Conservation of energy | [katex] E_i = E_f [/katex] |

Step #2. Identification of initial form of energy | [katex] E_i = \text{Work} [/katex](since the box is being pushed up) |

Step #3. identification of final form of energy | [katex] E_f = PE [/katex] (since the work energy being applied is causing the block to gain height, thus potential energy) |

Step #4. Substitute KE and PE into [katex] E_i = E_f [/katex] | [katex] E_f = E_i \rightarrow W = PE \rightarrow W = mgh [/katex] |

Step #5. Rearrange and solve of unknown | Solve for Work → [katex] W = mgh [/katex] = (.5)(9.8)(2) = 9.8 J |

## Show Further Explanation

As the block is pushed up the ramp, it gains gravitational potential energy due to its increasing height. The work done on the block by the applied force is equal to the change in the block’s gravitational potential energy, which is equal to the applied force times the distance the block moves.

*You might wonder why we couldn’t simply use W = Fd. It’s because we are given the *height * NOT the length (d) of the ramp. Remember that the distance and force applied must be parallel.

### Conservative vs Non-Conservative Systems

Let’s clear up a misconception: “Non-conservative” does NOT mean energy is not conserved in the system. [katex] E_i = E_f [/katex] (Energy is conserved) as long as no net external forces act on the system.

A **conservative force** does not depend on the path an object takes. Think about gravity acting on a block. Wether it’s sliding down a ramp, or being dropped straight down from the height of the ramp, it will still have the same energy.

A **non-conservative force** *does *depend on the path taken. Think about friction. If you push a block up or down a ramp, there will be energy “dissipated” that can’t be transformed back into potential or kinetic energy.

Non-Conservative Forces | Conservative Forces |

Friction and air resistance | Gravitational |

Tension | Elastic (springs, pendulums) |

Motors, engines, or rockets | Electric |

A person pushing/pulling |

Important things to remember:

- Work done by
*ALL*non-conservative forces, is just [katex] \text{Work} = Fd [/katex] - Another way to way to find the work done by a non-conservative force → [katex] W_{NC} = \Delta KE + \Delta PE [/katex]. This simply tells us that the amount of energy lost or gained from both potential and kinetic energies is the total amount of work done by a non-conservative force.
- If there is NO work done by non-conservative forces in a system, then the system has purely
*mechanical energy*. - HINT: use the problem solving framework above to solve any energy question, regardless of the types of forces in the system.

### Power

Power is a measure of the amount of energy dissipated over a period of time.

[katex] P = \frac{W}{t} [/katex]

Where “W” is the work done and “t” is the time taken.

The units are Joules/second. We commonly call this Watts (common in light bulbs) or Horsepower (common in cars).

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### Try these 10 Energy Questions

Both hit the ground at the same time.

Both hit the ground with the same speed.

The one thrown at an angle hits the ground with a lower speed.

The one thrown at an angle hits the ground with a higher speed.

Both (a) and (b)

*m*with constant speed

*v*across a rough horizontal floor. The coefficient of friction between the box and the floor is

*µ*. At what rate does the child do work on the box?

[katex] \mu mgv [/katex]

[katex] \mu mg/v [/katex]

[katex] \mu mg/v^2[/katex]

[katex] mgv [/katex]

Can not be determined.

Its potential energy is half of its initial potential energy.

Its speed is half of its initial speed.

Its total mechanical energy is half of its initial value.

Its kinetic energy is half of its initial kinetic energy.

None of the above.

The nonconservative force has a component that points opposite to the displacement of the object.

The nonconservative force is perpendicular to the displacement of the object.

The nonconservative force has a direction that is opposite to the displacement of the object.

The nonconservative force has a component that points in the same direction as the displacement of the object.

The nonconservative force points in the same direction as the displacement of the object.

They will traverse the distanced in the same time.

They will have the same velocity.

The work done on both vehicles is the same.

The average power expended is the same.

They will have the same kinetic energy