Worked examples — Equations of motion — 3DOF point mass (trajectory analysis)

Before we start, one reminder of the notation we will lean on (every symbol earned in the parent):

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The scenario matrix

Every 3DOF problem is really one of these cells. The examples below are labelled [C1], [C2], … so you can see each cell is covered.

Cell	Case class	What is special	Key term that changes sign / vanishes
C1	Steep climb, $0<\gamma <90^{\circ }$	thrust on, gravity slows & turns	$\sin \gamma >0,\cos \gamma >0$
C2	Vertical, $\gamma =90^{\circ }$	local accel, drag opposes climb	$\cos \gamma =0\Rightarrow \dot{\gamma }=0$
C3	Level flight, $\gamma =0^{\circ }$	gravity does not slow speed	$\sin \gamma =0,\cos \gamma =1$ (fastest turn)
C4	Descending / diving, $\gamma <0^{\circ }$	gravity now adds speed	$\sin \gamma <0\Rightarrow -g\sin \gamma >0$
C5	Coast / ballistic, $T=0$	engine off; drag $-D/m$ still slows, or $D=0$ ⇒ pure projectile	thrust term vanishes; drag term survives unless $D=0$
C6	Mass changing, $\dot{m}\ne 0$	same thrust, rising acceleration	$m$ shrinks over the burn
C7	Degenerate speed, $V\to 0$	turning rate blows up	$\dot{\gamma }=-g\cos \gamma /V\to \infty$
C8	Word problem (gravity turn)	initial pitch-over	small $\gamma$, uses full set
C9	Exam twist (lift added)	$L\ne 0$ curves path up	normal equation full form

We now hit each cell.

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C1 — Steep climb (all signs positive)

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C2 — Vertical flight ($\gamma =90^{\circ }$)

How to read figure s01. The horizontal axis is the flight-path angle $\gamma$ sweeping from level ($0^{\circ }$) to vertical ($90^{\circ }$); the vertical axis is a gravity acceleration in ${\text{m/s}}^2$. The amber curve is $g\sin \gamma$ — the slice of gravity that acts along the velocity and appears in the speed equation $\dot{V}$. The cyan curve is $g\cos \gamma$ — the slice that acts across the velocity and drives the turning equation $\dot{\gamma }$. Notice they cross at $45^{\circ }$ and trade places: at the left edge (level, C3) turning is maximal, at the right edge (vertical, C2) slowing is maximal. Each curve is literally the number you multiply by in the corresponding worked example.

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C3 — Level flight ($\gamma =0^{\circ }$): fastest turn

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C4 — Descending / diving ($\gamma <0^{\circ }$): gravity adds speed

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C5 — Coast / ballistic ($T=0$): recover the parabola

How to read figure s02. The axes are the ground frame: horizontal is downrange $x$ (m), vertical is altitude $h$ (m). The cyan curve is the coast trajectory — a perfect parabola. The amber arrows drawn along it are the velocity arrows at three instants; watch how their horizontal reach stays the same length while their vertical reach shrinks, tips over, then grows downward. That constant horizontal component is exactly the $\ddot{x}=0$ result we just verified — the picture and the algebra say the same thing.

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C6 — Mass changing ($\dot{m}\ne 0$): acceleration climbs

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C7 — Degenerate speed ($V\to 0$): turning rate blows up

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C8 — Word problem: the gravity-turn pitch-over

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C9 — Exam twist: lift bends the path up

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Recall

Recall Which term changes sign when the rocket dives?

The gravity term in the speed equation, $-g\sin \gamma$ ::: for $\gamma <0$, $\sin \gamma <0$ so $-g\sin \gamma >0$ — gravity adds speed in a dive.

Recall Why is

$\dot{\gamma }=0$ for a vertical rocket even at $V=0$? Because $\dot{\gamma }=-g\cos \gamma /V$ and $\cos 90^{\circ }=0$ ::: the numerator is zero, killing the blow-up; no sideways gravity means nothing turns the arrow.

Recall Same thrust, half the mass — what happens to

$\dot{V}$? It (roughly) doubles ::: $a=F/m$; shrinking $m$ raises acceleration, which is why burnout is violent.

Recall What is

$g_0$ and how is it different from $g$? $g_0=9.80665{\text{ m/s}}^2$ is a fixed reference constant used only inside $I_{sp}{g}_0$ to get exhaust speed ::: $g$ is the local gravity actually pulling the rocket and can vary with altitude; $g_0$ never changes.

Recall In a thrust-off coast, when does the

$-D/m$ term disappear? Only when $D=0$ (drag-free) ::: with $T=0$ the drag term still slows the vehicle unless drag itself is zero, which is the pure-projectile subcase that recovers the parabola.

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Connections

• Equations of motion — 3DOF point mass (trajectory analysis) — the parent set of rules these examples drill
• Tsiolkovsky Rocket Equation — the vertical, drag-free limit (C2)
• Gravity Turn Trajectory — the pitch-over of C7 and C8
• Drag and Atmospheric Models — supplies the $D$ used throughout
• Flight-Path Angle and Velocity Frame — geometry of every $\sin \gamma$/$\cos \gamma$ split
• 6DOF Rigid-Body Dynamics — where lift (C9) and rotation return
• Numerical Integration of Trajectories (RK4) — chaining these instantaneous rates into a full path