Foundations — Pitch program — open-loop pitch-over

Before you can read $\dot{\gamma }=-\frac{g\cos \gamma }{v}$, every letter in it must mean something you can see. This page builds each one from nothing, in the order they depend on each other.

0. The one picture everything hangs on

Everything in this topic happens in a flat vertical slice of the world: the ground is a horizontal line, "up" is the vertical line, and the rocket is a single dot moving through this slice. We ignore sideways-out-of-page motion entirely.

Why a point? Because for the trajectory shape we only care about where the dot goes, not how long the rocket is. Its shape matters only for drag and steering, which we handle separately.

1. The velocity arrow $\vec{v}$ and its speed $v$

The rocket is moving. At any instant, draw an arrow from the dot showing which way it goes and how fast.

Picture: a blue arrow leaving the dot. A long arrow = fast; a short arrow = slow. The tilt of this arrow is the next symbol.

Why the topic needs it: orbit is a demand on this arrow — it must eventually be long (~7.8 km/s) and flat (horizontal). The whole pitch program is about steering this one arrow.

2. The flight-path angle $\gamma$

Look at the velocity arrow. How steeply is it climbing? That tilt, measured up from the horizontal, is $\gamma$ ("gamma").

Why measure from horizontal (not vertical)? Because "orbit" means $\gamma =0$. Choosing horizontal as the zero makes the target a clean number, and the whole ascent is the story of $\gamma$ shrinking from $90^{\circ }$ down toward $0^{\circ }$.

3. Sine and cosine — splitting an arrow into up-and-along

To do physics with the tilted velocity arrow we must break it (and gravity) into a horizontal part and a vertical part. The tools that do this splitting are sine and cosine.

Take any arrow of length $L$ tilted at angle $\gamma$ above horizontal. Drop it into a right triangle:

Why these two and not something else? A right triangle has exactly two legs; $\cos$ hands you the one next to the angle (along the ground) and $\sin$ hands you the one across from it (up). They are the unique dial that converts "an arrow tilted by $\gamma$" into "how much of it is sideways vs upward."

Check the extremes:

• At $\gamma =90^{\circ }$ (straight up): $\cos 90^{\circ }=0$ (no horizontal), $\sin 90^{\circ }=1$ (all vertical). ✓
• At $\gamma =0^{\circ }$ (horizontal): $\cos 0^{\circ }=1$ (all horizontal), $\sin 0^{\circ }=0$ (no vertical). ✓

4. Gravity $g$ and its two components

$g$ is how hard the planet pulls each kilogram downward: $g\approx 9.8{\text{m/s}}^2$ near Earth's surface. It always points straight down in our picture.

But the velocity arrow is tilted, so gravity has two effects on the arrow: one that slows the climb (along $\vec{v}$) and one that bends the arrow (across $\vec{v}$).

Why split against the velocity arrow (not against the ground)? Because a force along the arrow changes only its length (speed), while a force across the arrow changes only its direction (angle). Splitting this way separates the two questions the topic asks: "how fast?" and "which way?"

5. Mass $m$, thrust $T$, drag $D$, and force

Picture: on the dot, a green arrow forward = thrust, a red arrow backward = drag, a yellow arrow down = weight $mg$.

Why mass appears everywhere: Newton says push = mass × (rate of change of velocity). More mass → the same push bends or speeds the arrow less. That is why $m$ cancels out of the turn law (it appears on both sides) — the turning of the direction doesn't care about mass, only the speeding up does.

6. The dot — "rate of change" notation

The parent writes $\dot{v}$ and $\dot{\gamma }$. The dot on top is a shorthand.

Picture: if $\dot{\gamma }$ is negative, the velocity arrow is tilting down (toward horizontal) as seconds tick by. The bigger the number, the faster the arrow swings.

Why the topic lives on $\dot{\gamma }$: the entire pitch program is a statement about how fast the arrow tilts over. $\dot{\gamma }<0$ is exactly "the nose is pitching over."

7. Radians — why angles get measured in a strange unit

$\dot{\gamma }$ came out in radians per second, not degrees. Radians are the "natural" way to measure angle when it multiplies lengths and speeds.

Why radians here? The turn equation $\dot{\gamma }=-g\cos \gamma /v$ mixes an angle-rate with speeds and accelerations. That algebra is only clean if the angle is in radians (so arc = angle × radius holds without extra factors). We convert to degrees only at the end for human intuition.

8. The thrust-to-weight ratio $n$

The parent sets $a=T/m$ and $n=a/g$. Build it up:

Picture: $n=1$ → engine exactly cancels weight (rocket hovers). $n=1.3$ → net upward push of $0.3g$; the rocket accelerates upward. See Thrust-to-weight ratio.

Why it matters: $n$ decides how quickly the speed grows, which (through $v$ in the turn law) decides how quickly the turn slows down. So $n$ and the kick angle together fix the whole ascent shape.

How these feed the topic

This whole chain terminates in the pitch program, and connects onward to the Gravity turn trajectory and, once feedback is added, Closed-loop ascent guidance (PEG / IGM).

Equipment checklist

Cover the right side and answer each before reading the parent note.