Terminal landing — propulsive descent, suicide burn: Foundations

Before you can trust the suicide-burn formula $h_{b u r n} = \frac{v ^{2}}{2 ( a _{T} - g )}$ , every letter in it must mean something you can picture. This page builds each symbol from nothing, in the order they depend on each other. Nothing is used before it is drawn.

0. The stage: a vertical number line

Everything happens along a single up–down line. We must first agree which direction is positive, or every sign later becomes a guess.

Figure — Terminal landing — propulsive descent, suicide burn

Why we need it: the rocket falls down but thrusts up. Without a fixed positive direction, "add" and "subtract" have no meaning. Look at the figure: the orange arrow (fall) and teal arrow (thrust) point opposite ways — the sign convention is what lets us combine them with $+$ and $-$ .

1. Altitude $h$ — how high

Picture: the vertical gap between the rocket and the floor. Why the topic needs it: the whole maneuver is a race between "height remaining" and "height needed to stop." $h$ is the height remaining.

2. Velocity $v$ — how fast, and which way

Picture: the length of the motion arrow; longer arrow = faster. The arrow points down during descent.

See Kinematics — Equations of Motion for the full velocity toolkit.

3. Acceleration $a$ — how fast the velocity changes

Picture: if velocity is the length of the motion arrow, acceleration is how fast that arrow is growing or shrinking each tick of the clock.

4. Gravity $g$ — the always-on downward pull

Picture: an invisible downward arrow attached to the rocket at all times, the same length whether the engine is on or off.

Why the topic needs it: this is the villain. Gravity keeps adding downward speed even while you brake, so your engine must beat gravity plus the fall. That is the single reason the formula has $a_{T} - g$ and not just $a_{T}$ .

5. Thrust $F$ and thrust acceleration $a_{T}$

Picture: the teal upward arrow in the figure below, opposing gravity. Why divide by mass? A given push moves a light rocket faster than a heavy one — that's Newton's second law, $a = F / m$ . See Thrust and Specific Impulse for where $F$ comes from.

6. Mass $m$ — how much stuff

Why the topic needs it: $m$ appears inside $a_{T} = F / m$ . As fuel burns, $m$ shrinks, so with steady thrust $a_{T}$ grows through the burn. The parent's constant- $m$ formula is a first estimate; the exact story lives in Tsiolkovsky Rocket Equation.

7. Net acceleration $a_{n e t}$ — the actual braking

Picture: in the figure above, $a_{n e t}$ is the leftover length of the teal thrust arrow after gravity's orange arrow cancels part of it.

The three cases every reader must see:

$a_{T} > g$ → net arrow points up → you decelerate → landing possible. ✓
$a_{T} = g$ → arrows cancel → net zero → you fall at constant speed forever → crash.
$a_{T} < g$ → net arrow points down → you keep speeding up → hopeless.

8. The kinematic tool $v^{2} = v_{i}^{2} + 2 a d$

Why THIS tool and not another? We want to relate speed and distance without caring about time (we don't ask "how many seconds to stop," we ask "how much height to stop"). This is the one equation of motion with no $t$ in it, so it answers exactly our question. Full derivation in Kinematics — Equations of Motion.

Applied here: enter braking at falling speed $v$ , want to reach $v_{f} = 0$ over height $h_{b u r n}$ , with braking acceleration $a_{n e t}$ : $0 = v^{2} - 2 a_{n e t} h_{b u r n} \Rightarrow h_{b u r n} = \frac{v ^{2}}{2 ( a _{T} - g )}$

The $v^{2}$ is why doubling your fall speed quadruples the room you need.

Up is positive, down is negative, ground is zero.

What does

h

mean and its unit?

Height above the ground, in metres; the "height remaining."

Difference between speed and velocity?

Speed is how fast; velocity is speed with a direction sign — falling is negative.

What does acceleration measure and its unit?

How much the velocity changes each second, in

m/s^{2}

What is

g

and does it ever turn off?

Downward gravitational acceleration (

\approx 10

on Earth,

1.6

on Moon); it never turns off.

Why is

a_{T} = F / m

Newton's second law — a given push

F

accelerates mass

m

F / m

Why does

a_{n e t} = a_{T} - g

Gravity still pulls during the burn, so only thrust above weight decelerates you.

What must be true of

a_{T}

g

for landing to work?

a_{T} > g

, so net acceleration points up.

Which kinematic equation relates speed and distance without time, and why use it?

v_{f}^{2} = v_{i}^{2} + 2 a d

; we want height-to-stop, not time-to-stop.

Fall speed from rest through height

H

v = 2 g H

Connections

Parent: suicide burn
Kinematics — Equations of Motion — where $v^{2} = 2 a d$ and $v = 2 g H$ come from
Thrust and Specific Impulse — where $a_{T} = F / m$ comes from
Tsiolkovsky Rocket Equation — what happens when $m$ changes

Foundations — Terminal landing — propulsive descent, suicide burn