Visual walkthrough — Orbital velocity for circular orbit — derivation
Step 1 — Why a fast sideways throw becomes an orbit
WHAT. Imagine standing on a very tall mountain and throwing a ball straight sideways. Gravity pulls it down; it lands. Throw harder — it lands further away. Throw insanely hard, and something strange happens.
WHY. Before any algebra, we need the physical picture: an orbit is nothing mystical — it is a projectile thrown so fast that the ground curves away beneath it exactly as fast as it falls. This is the whole idea, and every symbol later is just this picture made precise.
PICTURE. Look at the figure. The three black arcs are gentle, medium, and hard throws — they all curve down and hit the ground. The red curve is the special throw: it keeps missing the ground forever and closes into a circle.
Step 2 — Turning in a circle is an acceleration
WHAT. Even though the special red ball moves at a constant speed, its direction keeps changing. A change in the velocity arrow — even just its direction — is an acceleration.
WHY. We introduce acceleration here because Newton's 2nd law () only bites when there is an acceleration. So we must first prove the orbiting ball is accelerating, and find which way.
PICTURE. The figure shows the velocity arrow at two nearby moments (black). Slide the second arrow back onto the first: the tiny red arrow that bridges their tips points straight toward the centre. That inward direction is the direction of the acceleration.
Step 3 — Something must supply that inward force
WHAT. Newton's 2nd law says: to accelerate a mass inward at , a real inward force must exist of size .
WHY. Acceleration never happens on its own — a force causes it. So we name the force the circle demands. We do not yet know what provides it; we only know how big it must be.
PICTURE. The red arrow on the ball points inward, labelled . It is the "bill" the circle sends us: "give me this much inward force or I stop being a circle."
Step 4 — The Earth pays the bill with gravity
WHAT. Now we ask: what real force is available? The only force on a satellite in vacuum (no engine, no air) is Earth's gravity, pulling it inward.
WHY. We must connect the demand (Step 3) to a real supply. Newton's law of gravitation gives the exact size of that inward pull. See Newton's Law of Universal Gravitation.
PICTURE. The figure keeps the same ball but now the inward red arrow is labelled — the gravitational pull from Earth's centre. Notice it points the same way as the arrow of Step 3. That alignment is the whole secret.
Step 5 — Gravity is the centripetal force (the key claim)
WHAT. We are not adding two forces. Gravity is the only force, and it happens to point exactly where the circle needs its centripetal force. So gravity plays the role of the centripetal force: set them equal.
WHY. This single equality is the physical heart of the whole derivation. Every earlier step existed to justify writing this one line.
PICTURE. The two red arrows from Steps 3 and 4 are drawn on top of each other — same length, same direction. When "the bill" and "the payment" match perfectly, the orbit is stable.
Step 6 — Cancel, and watch the mass disappear
WHAT. Both sides have a factor of (the ball's mass) and both have an . Cancel them.
WHY. Cancelling is not just tidying — the vanishing of carries a deep truth: the orbit does not care how heavy the satellite is.
PICTURE. The figure shows a feather and a truck on the same red orbit at the same radius, moving at the same speed. The mass literally strikes through and cancels.
Step 7 — The law, drawn
WHAT. The formula says: as grows, shrinks — but slowly, like a square-root, not straight down.
WHY. Numbers are abstract; a curve is memorable. Seeing the shape lets you predict without computing: quadruple , halve .
PICTURE. The red curve falls steeply near small , then flattens for large . Two marked points show that going from to drops the speed to exactly half.
Step 8 — The degenerate / edge cases
Every scenario must be covered so you never hit a surprise.
WHAT & PICTURE. The figure stacks three cases as red markers on the curve:
The one-picture summary
The whole story in a single frame: the circle demands an inward force (), gravity supplies it (), setting them equal and cancelling and one yields — with the red formula glowing at the centre of the orbit.
Recall Feynman retelling — the walkthrough in plain words
Throw a ball sideways off a mountain. Gravity always drags it down, so it curves and lands. Throw harder and it lands further. Throw it at just the right insane speed — about 8 kilometres every second — and here's the trick: the round Earth curves away below it exactly as fast as it falls, so it keeps missing the ground and loops around forever. That's an orbit — falling sideways so fast you never land.
Now why that exact speed? Going in a circle means your direction keeps turning, and turning is a kind of acceleration pointing toward the centre — its size is . Newton says acceleration needs a force, so the circle demands an inward force . The only thing pushing our ball inward is gravity, which pulls with . Since gravity is the only force and it points exactly inward, gravity is that centripetal force — so we set the two equal. When we do, the ball's own mass cancels off both sides (that's why a feather and a truck orbit the same), one cancels too, and we're left with . Lower orbits are faster, far-out orbits are lazy, and you can never orbit the centre itself.
Recall Quick self-test
- Which way does the acceleration point in a circle? ⇒ Toward the centre (centripetal).
- What supplies the centripetal force for a satellite? ⇒ Gravity, and gravity alone.
- Where does the go in the final formula? ⇒ It cancels — orbit is mass-independent.
- What is for a satellite km up? ⇒ km, from Earth's centre.
- Speed at compared to ? ⇒ Half, because .
Connections
- Parent topic — full derivation
- 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