Gravity and Orbits

Part 2: The Physics of Motion

Vectors and Key Variables: The Language of Motion

Before diving into Newton’s laws, we need to establish notation. Physics uses symbols as shorthand — but those symbols must be defined before we use them.

The key variables we’ll use:

The arrow notation ($\vec{v}$ vs. $v$) distinguishes vectors from scalars. When we write $v$ without the arrow, we mean the speed — just the magnitude (always zero or positive). Velocity components like $v_x$ can be positive or negative: the sign tells you which way that part of the motion points.

Setting the Stage: Newton’s Laws of Motion

Before Newton could explain why planets move as they do, he needed the fundamental rules of motion itself.

First Law (Inertia):

An object remains at rest or in uniform motion unless acted upon by a net external force.

Mathematically: if $\sum \vec{F} = 0$, then $\vec{v} = \text{constant}$. Motion doesn’t require continuous effort — only changes in motion require forces.

Second Law (Force and Acceleration):

The net force on an object equals its mass times its acceleration.

$\vec{F} = m\vec{a}$

This is the heart of Newtonian mechanics: forces cause accelerations, mediated by mass.

For the same force, a more massive object accelerates less — this resistance is inertia. Unit check: $[F] = [m][a] = \mathrm{g} \times \mathrm{cm/s^2} = \mathrm{g\,cm/s^2} = \mathrm{dyne}$ ✓.

Third Law (Action-Reaction):

For every action there is an equal and opposite reaction: $\vec{F}_{AB} = -\vec{F}_{BA}$.

If the Sun pulls on Earth, Earth pulls back on the Sun with exactly the same force. This matters for center of mass and binary systems.

Kinematics: Describing Motion Quantitatively

To apply Newton’s laws, we need precise language for describing motion — position, velocity, and acceleration.

Velocity ($\vec{v}$): how position changes with time, both speed and direction.

$\vec{v} = \frac{d\vec{r}}{dt} \approx \frac{\Delta \vec{r}}{\Delta t} \quad \text{(for small } \Delta t\text{)}$

Units: $[v] = \mathrm{cm/s}$.

Acceleration ($\vec{a}$): how velocity changes with time.

$\vec{a} = \frac{d\vec{v}}{dt} \approx \frac{\Delta \vec{v}}{\Delta t} \quad \text{(for small } \Delta t\text{)}$

Units: $[a] = \mathrm{cm/s^2}$. Acceleration doesn’t just mean “speeding up” — it means any change in velocity, including slowing down or changing direction. A planet in a circular orbit at constant speed is still accelerating because its direction continuously changes. Your car’s speedometer can stay at 60 mph while you round a curve, but you still have to turn the wheel — that steering effort is the acceleration.

Circular Motion and the Centripetal Force

Here’s the key insight connecting motion physics to orbits: an object moving in a circle is constantly accelerating, even at constant speed. Because velocity is a vector, and for circular motion its direction is always changing.

Faster motion means direction changes more rapidly; a tighter curve (smaller radius) means direction changes more sharply. These combine: for an object moving in a circle of radius $r$ with speed $v$, the centripetal acceleration points toward the center with magnitude

$a_c = \frac{v^2}{r}$

Unit check: $[a_c] = (\mathrm{cm/s})^2 / \mathrm{cm} = \mathrm{cm/s^2}$ ✓. By Newton’s second law, this acceleration requires a force — the centripetal force:

$F_c = ma_c = \frac{mv^2}{r}$

This isn’t a new type of force — it’s whatever force happens to be pulling the object toward the center. For planets, that force is gravity; for a ball on a string, tension; for a car turning, friction.