Foundations — Orbital velocity for circular orbit — derivation

Before you can enjoy the derivation on the parent page, the topic note, you must recognise every letter and every picture it silently relies on. This page builds each one from nothing, in the order that each new piece leans on the previous.

1. What a "vector" and "speed" mean here

Look at the amber arrow in the figure. Its length is the speed; its pointing is the direction. As the satellite rides around the circle, the arrow stays the same length but swings around — that swinging is exactly what "accelerating" will mean in a moment.

2. The circle: radius $r$, and where we measure it

In the figure the cyan line $R$ runs from the planet's centre to its surface (the planet's own radius). The white line $h$ is the altitude — how high above the ground the satellite floats. The full radius is the two added together:

$r=R+h$

Why do we even need $r$? Because both the strength of gravity and the sharpness of the turn depend on how far out you are. It is the single most important dial in the whole topic.

3. Acceleration — the arrow that points inward

The size of this inward acceleration has a name and a formula we will simply use (its full derivation lives in Centripetal Force and Uniform Circular Motion):

4. Force and mass — Newton's Second Law

So the inward force needed to keep a mass $m$ turning is:

$F_c=m{a}_c=\frac{m{v}^2}{r}$

5. Gravity — the pull that does the bending

Let us earn each symbol:

• $G$ — the gravitational constant, a fixed number of nature ($6.67\times 10^{-11}$ in SI units). It sets the overall strength of gravity everywhere in the universe.
• $M$ — the big mass (the planet). Bigger $M$ $\Rightarrow$ stronger pull.
• $m$ — the small mass (the satellite).
• $r^2$ — the distance squared in the bottom. This is the inverse-square law: go twice as far, gravity drops to a quarter.

6. Surface gravity $g$ and the shortcut $GM=g{R}^2$

7. The square root  ​ — undoing a square

The derivation ends with $v_o^2=\frac{GM}{r}$. We have $v_o^2$ but want $v_o$, so we take the square root of both sides to "peel off the square." That is the only reason the root appears — it is the tool that undoes the squaring that circular motion ($v^2$) forced on us.

Prerequisite map

Equipment checklist

Test yourself — you are ready for the derivation when you can answer each without peeking.

Connections

• Orbital velocity for circular orbit — derivation — the parent this page prepares you for.
• Centripetal Force and Uniform Circular Motion
• Newton's Law of Universal Gravitation
• Acceleration due to gravity g and GM = gR²
• Escape Velocity
• Kepler's Third Law
• Geostationary Orbit