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

This page assumes you have seen none of the notation on the parent page. We build each symbol from a picture, in an order where each one only needs the ones before it. By the end you can read the parent equations left to right without ever meeting an unexplained mark.

0. The picture we live inside

Everything happens in a flat vertical plane: a floor (the ground) and a wall (straight up). We watch one dot — the rocket, shrunk to a point — move across this plane.

1. Position: where the dot is — $x$ and $h$

To say where the dot is, we need two numbers, because the plane is two-dimensional.

Why the topic needs them: the final product of a trajectory analysis is a curve of $h$ against $x$ — the flight path drawn on the wall. Everything else exists to compute how $(x,h)$ changes.

2. The velocity arrow: $V$ and $\gamma$

The dot moves, so it has a velocity — an arrow whose length is the speed and whose direction is where it's heading. Instead of splitting velocity into "horizontal part" and "vertical part," the parent note describes the arrow directly with its length and its tilt. This is the single most important idea to get comfortable with.

3. Reading an angle: sine and cosine on the velocity triangle

The symbols $\sin \gamma$ and $\cos \gamma$ appear everywhere in the parent note. Here is what they are, from zero.

Drop the velocity arrow into a right triangle: its horizontal shadow, its vertical rise, and the arrow itself as the slanted side (the hypotenuse, length $V$).

This immediately explains two boxed kinematic equations from the parent: $\dot{x}=V\cos \gamma (\text{horizontal part}),\dot{h}=V\sin \gamma (\text{vertical part}).$

4. The dot on top: $\dot{V}$, $\dot{\gamma }$, $\dot{x}$, $\dot{h}$, $\dot{m}$

The parent note is full of symbols wearing a dot on top. This is not decoration.

An equation with a dotted variable on the left ("$\dot{V}=\dots$") is called an ODE — an equation for a rate. The parent's four boxed equations are four such rate-rules.

5. The four forces and their symbols

The rate of the arrow's length and tilt is set by forces. Four of them, each with its own symbol.

6. The last two symbols: $I_{sp}$ and $g_0$

In the mass-rate rule $\dot{m}=-T/(I_{sp}{g}_0)$ two new symbols appear.

Prerequisite map

Read it as: the dot gives us position and velocity; velocity gives length ($V$) and tilt ($\gamma$); trig splits the arrow; forces split by trig set the rates; all rates together are the 3DOF equations. See also Flight-Path Angle and Velocity Frame for the geometry of $\gamma$, and 6DOF Rigid-Body Dynamics for what we deliberately left out.

Equipment checklist

Cover the answers and test yourself before opening the parent note.