# Motion in Gravitational Fields

> Newton's law of gravitation, orbits, Kepler's laws, satellites, gravitational potential energy and escape velocity.

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

### Story time: Newton's cannon
Isaac Newton imagined a cannon on top of a mountain so tall it pokes out of the air. Fire a ball gently and it curves down and lands nearby. Fire it faster and it lands further away.

Fire it fast enough, and something strange happens. The ball falls towards Earth, but Earth's surface curves away beneath it just as fast. The ball keeps falling and keeps missing. It goes all the way round and comes back to the mountain.

The ball is in **orbit**. Every satellite, the Moon, and Earth itself around the Sun are doing exactly this: falling forever, and never landing.

**In short:** Every mass attracts every other mass with a force $F = \dfrac{GMm}{r^2}$, so doubling the distance quarters the force. For a satellite in a circular orbit, gravity is the centripetal force, which sets its speed: $v = \sqrt{\dfrac{GM}{r}}$, so higher orbits are slower.

## Newton's law of universal gravitation

Every mass attracts every other mass. The force between two masses $M$ and $m$, with centres a distance $r$ apart, is:

$$F = \frac{GMm}{r^2}$$

- $G = 6.67 \times 10^{-11}$ N m² kg⁻² (the gravitational constant)
- $r$ is measured **centre to centre**, not from the surface.

The two masses pull on each other with **equal and opposite** forces (Newton's third law). Earth pulls on you exactly as hard as you pull on Earth.

This is an **inverse square law**:

| Distance | Force |
|---|---|
| $r$ | $F$ |
| $2r$ | $F/4$ |
| $3r$ | $F/9$ |

## Gravitational field strength

The **gravitational field strength** $g$ is the force on each kilogram:

$$g = \frac{GM}{r^2}$$

At Earth's surface, $g = 9.8$ N/kg. At twice Earth's radius from the centre, it is a quarter of that: 2.45 N/kg.

### Worked example: Gravity on Mars
Mars has mass $6.4 \times 10^{23}$ kg and radius $3.4 \times 10^6$ m.

$$g = \frac{6.67 \times 10^{-11} \times 6.4 \times 10^{23}}{(3.4 \times 10^6)^2} = 3.7 \text{ N/kg}$$

## Orbits: gravity is the centripetal force

For a satellite in a **circular** orbit, gravity provides the centripetal force:

$$\frac{GMm}{r^2} = \frac{mv^2}{r} \quad\Rightarrow\quad v = \sqrt{\frac{GM}{r}}$$

- The satellite's own mass cancels out. A tiny satellite and a huge one at the same height move at the same speed.
- **Higher** orbits are **slower** and take **longer**.

Using $v = \dfrac{2\pi r}{T}$, the period is $T = 2\pi\sqrt{\dfrac{r^3}{GM}}$.

## Kepler's three laws

Johannes Kepler found three rules by studying careful records of the planets.

1. **Orbits are ellipses**, with the Sun at one **focus**.
2. A line from the Sun to a planet sweeps out **equal areas in equal times**. So a planet moves **fastest** when it is **closest** to the Sun.
3. For every planet around the same star, $\dfrac{r^3}{T^2}$ is the same:

$$\frac{r^3}{T^2} = \frac{GM}{4\pi^2}$$

This third law lets you find the mass of a star or planet from the orbits around it.

### Worked example: Comparing two moons
Moon A orbits a planet at radius $r$ with period 2 days. Moon B orbits at $4r$. Its period is:

$$T_B = T_A \times \left(\frac{4r}{r}\right)^{3/2} = 2 \times 8 = 16 \text{ days}$$

## Types of satellite orbits

| | Low Earth orbit (LEO) | Geostationary orbit |
|---|---|---|
| Height | about 200 to 2000 km | about 36 000 km |
| Period | about 90 minutes | 24 hours (one Earth day) |
| Speed | faster (about 7.8 km/s) | slower (about 3.1 km/s) |
| Uses | photos, weather, ISS | TV, communications |
| Notes | close to the ground, so detailed images | stays above one spot on the **equator** |

A geostationary satellite turns with Earth, so a dish on the ground can point at it all the time.

## Energy in a gravitational field

Far from a planet, we take gravitational potential energy as **zero**. Closer in, it is **negative**:

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

- The closer you are, the more negative $U$ is.
- Moving **away** means $U$ **increases** (gets closer to zero). This takes work.
- Near the surface, $\Delta U = mg\Delta h$ is a good shortcut, because $g$ hardly changes over small heights.

## Escape velocity

The **escape velocity** is the launch speed that lets an object get infinitely far away and just stop. Its total energy is then zero:

$$\frac{1}{2}mv_{esc}^2 - \frac{GMm}{r} = 0 \quad\Rightarrow\quad v_{esc} = \sqrt{\frac{2GM}{r}}$$

For Earth's surface, this is about 11.2 km/s. It does **not** depend on the mass of the object being launched.

### Watch out: "There is no gravity in space"
Astronauts on the ISS float, but gravity there is still about 90% of its value on the ground. They float because they and the station are **falling together** in orbit.

**Mind map**

```text
Gravitational Fields
  Newton's law
    F = GMm / r²
    Inverse square
    Equal and opposite forces
  Field strength
    g = GM / r²
    9.8 N/kg at Earth's surface
  Orbits
    Gravity = centripetal force
    v = √(GM / r)
    Higher is slower
  Kepler
    Ellipses
    Equal areas, equal times
    r³ / T² constant
  Satellites
    Low Earth orbit
    Geostationary, 24 h
  Energy
    U = −GMm / r
    E = −GMm / 2r
    Escape v = √(2GM / r)
```

### Year 11: Where g = 9.8 comes from
Put Earth's mass ($6.0 \times 10^{24}$ kg) and radius ($6.371 \times 10^6$ m) into $g = \dfrac{GM}{r^2}$ and you get 9.86 N/kg. On the Moon, the smaller mass gives about 1.6 N/kg.

A planet with twice Earth's mass and twice its radius has $g = \dfrac{G(2M)}{(2r)^2} = \dfrac{1}{2} \times 9.8 = 4.9$ N/kg. Always compare using the formula, not guesses.

### Year 12: Total energy of an orbit
For a circular orbit, the kinetic energy is $K = \frac{1}{2}mv^2 = \dfrac{GMm}{2r}$. The total energy is:

$$E = K + U = \frac{GMm}{2r} - \frac{GMm}{r} = -\frac{GMm}{2r}$$

So $K = -\frac{1}{2}U$ and $E = \frac{1}{2}U$. The total energy is negative: the satellite is **bound** to the planet.

**Changing orbit.** Moving to a higher orbit makes $E$ less negative, so it needs energy from rockets. In the new orbit, $U$ is higher but $K$ is lower: the satellite moves more slowly.

**Orbital decay.** In low orbit, thin air drags on a satellite. It loses total energy and drops lower. Strangely, it **speeds up**, because $v = \sqrt{GM/r}$ grows as $r$ shrinks. Its potential energy falls by twice as much as its kinetic energy rises.

**Area under a graph.** On a graph of gravitational force against distance, the area between two distances is the **work done**, which equals the change in potential energy.

**Only gravity acts.** A satellite in orbit has just one force on it, gravity, which acts as the centripetal force. There is no forward "thrust" force needed to keep it moving.

**Escape from an orbit.** Since $v_{\text{orbit}} = \sqrt{\dfrac{GM}{r}}$ and $v_{\text{esc}} = \sqrt{\dfrac{2GM}{r}}$, the escape velocity from any orbit radius is $\sqrt{2}$ times the orbital speed there. A satellite orbiting at 2100 m/s needs about 2970 m/s to escape.

**Work between two distances.** $W = \Delta U = GMm\left(\dfrac{1}{r_1} - \dfrac{1}{r_2}\right)$. Moving from $r$ to $3r$ takes $\frac{2}{3}$ of the energy needed to escape from $r$; moving on from $3r$ to $9r$ takes a third as much again as the first step, because the bracket $\frac{1}{3r} - \frac{1}{9r} = \frac{2}{9r}$ is a third of $\frac{1}{r} - \frac{1}{3r} = \frac{2}{3r}$.

**Graphs.** A graph of $g$ against $\dfrac{1}{r^2}$ is a straight line through the origin with gradient $GM$, so $M = \dfrac{\text{gradient}}{G}$.

**Where the pulls cancel.** Between Earth (mass $81M$) and the Moon (mass $M$), the pulls are equal where $\dfrac{81M}{x^2} = \dfrac{M}{(d - x)^2}$. Taking square roots, $9(d - x) = x$, so $x = \frac{9}{10}d$ from Earth's centre.

## Quick summary

- $F = \dfrac{GMm}{r^2}$, with $r$ centre to centre. Doubling $r$ quarters the force.
- $g = \dfrac{GM}{r^2}$ is the force per kilogram.
- In a circular orbit, gravity is the centripetal force, so $v = \sqrt{\dfrac{GM}{r}}$. Higher orbits are slower.
- Kepler: ellipses, equal areas in equal times, and $\dfrac{r^3}{T^2}$ constant for one central body.
- Geostationary satellites orbit above the equator once every 24 hours.
- $U = -\dfrac{GMm}{r}$, and escape velocity is $\sqrt{\dfrac{2GM}{r}}$.

## Quiz

Try these questions from the chapter quiz.

1. Which law says the gravitational force between two masses falls as 1/r²?
   - A. Kepler's first law
   - B. Newton's law of universal gravitation
   - C. Newton's first law
   - D. Kepler's second law
   - Answer: B. Newton's law of universal gravitation. F = GMm/r² is Newton's law of universal gravitation.

2. The distance between two moons doubles. The gravitational force between them becomes:
   - A. Twice as big
   - B. Half as big
   - C. One quarter as big
   - D. Four times as big
   - Answer: C. One quarter as big. F is proportional to 1/r². (1/2)² = 1/4.

3. Earth pulls on the Moon with a force F. The Moon pulls on Earth with a force of:
   - A. Much less than F
   - B. Exactly F
   - C. Much more than F
   - D. Zero
   - Answer: B. Exactly F. Newton's third law: the forces are equal in size and opposite in direction.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/physics/gravitational-motion/#quiz