# Potential and Kinetic Energy

## Energy

Energy is the capacity to do **work**.

The unit of energy is **J** (Joule) which is also **kg m ^{2}/s^{2}**

Energy can be in many forms! Here we look at Potential Energy (PE) and Kinetic Energy (KE).

## Potential Energy and Kinetic Energy

A hammer:

- when raised up has
**potential**energy (the energy of position or state) - when falling down has
**kinetic**energy (the energy of motion)

### Potential energy (PE) is **stored energy** due to position or state

- a raised hammer has PE due to gravity.
- fuel and explosives have Chemical PE
- a coiled spring or a drawn bow also have PE due to their state

### Kinetic energy (KE) is energy of **motion**

A moving car has a lot of **kinetic energy**

### From PE to KE

These skydivers have **potential energy** due to being high up.

After they jump this potential energy gets

converted into **kinetic energy** (and heat) as they speed up.

## Gravitational Potential Energy

When the PE is due to an objects height then:

PE due to gravity = m g h

Where:

- m is the objects mass (kg)
- g is the "gravitational field strength" of
**9.8 m/s**near the Earth's surface^{2} - h is height (m)

### Example: This 2 kg hammer is 0.4 m up. What is it's PE?

PE = m g h

PE = 2 kg × 9.8 m/s^{2} × 0.4 m

PE = 7.84 kg m^{2}/s^{2} = **7.84 J**

## Kinetic Energy

The formula is:

KE = ½ m v^{2}

Where

- m is the object's mass (kg)
- v is the object's speed (m/s)

### Example: What is the KE of a 1500 kg car going at suburban speed of **14 m/s** (about 50 km/h or 30 mph)?

KE = ½ m v^{2}

KE = ½ × 1500 kg × (14 m/s)^{2}

KE = 147,000 kg m^{2}/s^{2}

KE = **147 kJ**

Let's double the speed!

### Example: The same car is now going at highway speed of **28 m/s** (about 100 km/h or 60 mph)?

KE = ½ m v^{2}

KE = ½ × 1500 kg × (28 m/s)^{2}

KE = 588,000 kg m^{2}/s^{2}

KE = 588 kJ

Wow! that is a big increase in energy! Highway speed is way more dangerous.

**Double** the speed and the KE increases by **four** times. Very important to know

### A 1 kg meteorite strikes the Moon at 11 km/s. How much KE is that?

KE = ½ m v^{2}

KE = ½ × 1 kg × (11,000 m/s)^{2}

KE = 60,500,000 J

KE = 60.5 MJ

That is 100 times the energy of a car going at highway speed.

## From PE to KE

When falling, an object's **PE due to gravity** converts into **KE** and also **heat** due to air resistance.

Let's drop something!

### Example: We drop this 0.1 kg apple 1 m. What speed does it hit the ground with?

At 1 m above the ground it's Potential Energy is

PE = m g h

PE = 0.1 kg × 9.8 m/s^{2} × 1 m

PE = 0.98 kg m^{2}/s^{2}

Ignoring air resistance (which is small for this little drop anyway) that PE gets converted into KE:

KE = ½ m v^{2}

Swap sides and rearrange:

½ m v^{2} = KE

v^{2} = 2 × KE / m

v = √( 2 × KE / m )

Now put PE into KE and we get:

v = √( 2 × 0.98 kg m^{2}/s^{2} / 0.1 kg )

v = √( 19.6 m^{2}/s^{2} )

v = 4.427... m/s

Note: we can combine the formulas like this:

Velocity from KE: | v = √( 2 × KE / m ) | |

Put in formula for PE: | v = √( 2 × mgh / m ) | |

Cancel m/m: | v = √( 2gh ) |

**The mass does not matter!** It is all about height and gravity. For our earlier example:

v = √( 2gh )

v = √( 2 × 9.8 m/s^{2} × 1 m )

v = 4.427... m/s

## Summary

- Energy is the ability to do work
- Potential Energy (PE) is
**stored energy**due to position or statePE due to gravity = m g h

- Kinetic Energy (KE) is energy of
**motion**KE = ½ m v

^{2}