# Kinetic and Potential Energy

> Calculating the energy of moving and raised objects, and swapping between the two.

From HSC Physics on WhizzNotes, free study notes for NSW students: https://whizznotes.com/physics/kinetic-and-potential-energy/. Updated 2 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The roller coaster engineer
Aisha is designing a roller coaster for a theme park. The first hill is 40 m tall. A chain lifts the cars slowly to the top. Then, from there on, there is no motor at all.

Her boss asks: "How fast will the cars be going at the bottom? And how high can the second hill be?"

Aisha does not need to build anything to answer. She only needs two formulas: one for the energy stored by height, and one for the energy of motion. Energy at the top must equal energy at the bottom.

**In short:** Kinetic energy is the energy of a moving object, $E_k = \tfrac{1}{2}mv^2$. Gravitational potential energy is the energy an object has because of its height, $E_p = mgh$. As an object falls, its potential energy turns into kinetic energy.

## Kinetic energy

**Kinetic energy** ($E_k$) is the energy an object has because it is **moving**.

$$E_k = \frac{1}{2}mv^2$$

- $m$ = mass (kg)
- $v$ = speed (m/s)
- $E_k$ = kinetic energy (J)

### Key idea: Speed matters most
Speed is **squared**. So:
- Double the mass: $E_k$ doubles.
- Double the speed: $E_k$ becomes **four times** bigger.

This is why a car crash at 100 km/h is so much worse than one at 50 km/h. There is four times the energy to get rid of.

### Worked example: A running dog
Biscuit the dog has a mass of 20 kg and runs at 5 m/s.

$$E_k = \tfrac{1}{2} \times 20 \times 5^2 = \tfrac{1}{2} \times 20 \times 25 = 250 \text{ J}$$

### Watch out: Square only the speed
In $\tfrac{1}{2}mv^2$, only $v$ is squared. Do $v^2$ first, then multiply by $m$, then halve.

## Gravitational potential energy

**Gravitational potential energy** ($E_p$ or GPE) is energy stored because an object is **raised up** against gravity.

$$E_p = mgh$$

- $m$ = mass (kg)
- $g$ = gravitational field strength (9.8 N/kg on Earth)
- $h$ = height raised (m)

### Worked example: A book on a shelf
A 2 kg book is lifted onto a shelf 1.5 m high.

$$E_p = 2 \times 9.8 \times 1.5 = 29.4 \text{ J}$$

That 29.4 J came from the energy in your muscles. If the book falls, it turns back into kinetic energy.

## Swapping between kinetic and potential

When something falls (ignoring air resistance), **all the GPE it loses becomes kinetic energy**:

$$mgh = \tfrac{1}{2}mv^2$$

The mass cancels! So the speed at the bottom is:

$$v = \sqrt{2gh}$$

### Worked example: Aisha's roller coaster
Top of the 40 m hill, nearly at rest. Speed at the bottom (no friction):

$$v = \sqrt{2 \times 9.8 \times 40} = \sqrt{784} = 28 \text{ m/s}$$

That is about 100 km/h. And the second hill must be **lower** than 40 m, because some energy is always lost to friction and air resistance. Without extra energy, the cars could never climb higher than where they started.

## A swinging pendulum

- At the **highest** points: most GPE, zero kinetic energy (it stops for an instant).
- At the **lowest** point: least GPE, most kinetic energy (fastest).
- Over time, air resistance turns some energy into heat, so each swing gets a bit lower.

**Mind map**

```text
Mechanical energy
  Kinetic
    Ek = ½mv²
    Moving objects
    Double v, four times Ek
  Gravitational potential
    Ep = mgh
    Raised objects
    g = 9.8 N/kg
  Swapping
    Falling: Ep to Ek
    Rising: Ek to Ep
    v = √(2gh)
  Losses
    Friction
    Air resistance
    Becomes heat
```

### Year 11: Elastic potential energy and the work-energy theorem
A spring stretched by extension $x$ obeys **Hooke's law**, $F = kx$, where $k$ is the spring constant (N/m). The energy stored is:

$$E_{\text{elastic}} = \tfrac{1}{2}kx^2$$

This equals the area under a force-extension graph.

The **work-energy theorem**: the net work done on an object equals its change in kinetic energy.

$$W_{\text{net}} = \Delta E_k = \tfrac{1}{2}mv^2 - \tfrac{1}{2}mu^2$$

Example: a 1000 kg car braking from 20 m/s to rest loses $\tfrac{1}{2}(1000)(20)^2 = 200\,000$ J. If the braking force is 8000 N, it stops in $200\,000 \div 8000 = 25$ m.

### Year 12: Gravitational potential energy far from Earth
$E_p = mgh$ only works near the surface, where $g$ is constant. For orbits and space travel, use:

$$U = -\frac{GMm}{r}$$

It is negative because zero is set at infinite distance, and you have to **add** energy to escape. Setting kinetic energy equal to the energy needed to escape gives the **escape velocity**:

$$v_{\text{esc}} = \sqrt{\frac{2GM}{r}}$$

For Earth this is about 11.2 km/s.

## Quick summary

- Kinetic energy: $E_k = \tfrac{1}{2}mv^2$. Double the speed gives four times the energy.
- Gravitational potential energy: $E_p = mgh$.
- Falling objects turn $E_p$ into $E_k$. Rising objects do the reverse.
- With no friction, $mgh = \tfrac{1}{2}mv^2$, so $v = \sqrt{2gh}$, whatever the mass.
- In real life, friction and air resistance turn some energy into heat.

## Quiz

Try these questions from the chapter quiz.

1. A 4 kg ball moves at 3 m/s. Its kinetic energy is:
   - A. 6 J
   - B. 12 J
   - C. 18 J
   - D. 36 J
   - Answer: C. 18 J. Ek = ½ × 4 × 3² = ½ × 4 × 9 = 18 J.

2. A 50 kg student climbs stairs 4 m high. Using g = 10 N/kg, her gain in GPE is:
   - A. 200 J
   - B. 2000 J
   - C. 12.5 J
   - D. 500 J
   - Answer: B. 2000 J. Ep = mgh = 50 × 10 × 4 = 2000 J.

3. If the speed of a car doubles, its kinetic energy:
   - A. Doubles
   - B. Halves
   - C. Becomes four times bigger
   - D. Stays the same
   - Answer: C. Becomes four times bigger. Ek depends on v². 2² = 4.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/physics/kinetic-and-potential-energy/#quiz