Foundations — Tsiolkovsky rocket equation — full first-principles derivation from momentum

This is the D1 Foundations page for the parent topic. Read it before the derivation. We assume you have seen nothing — not even the arrow on top of a vector. Every symbol used in the main note is built here, in an order where each one leans only on the ones before it.

0. What a symbol even is here

A symbol is a short name for a number that describes something physical. When we write $m$, we mean "the amount of stuff in the rocket right now" — a single number, in kilograms. Nothing more mysterious than a label on a box. Our job below is to make sure every label is attached to a real picture before it appears in an equation.

1. Mass, $m$ — "how much stuff"

Picture a bathroom scale. The number it shows is (proportional to) $m$. For a rocket, $m$ is the rocket's shell plus whatever fuel is still on board at this instant.

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

Look at the figure: the rocket moves right along one straight line. The amber arrow is $v$. A longer arrow means faster; if the arrow flipped to point left, $v$ would be negative. Because everything happens along a single line, we can drop the arrow-on-top notation and treat $v$ as a plain signed number.

3. Change and the symbol $\Delta$ (delta)

Picture two snapshots: before and after. If the rocket started at $v_i=0$ and ended at $v_f=4000$ m/s, then $\Delta v=4000-0=4000$ m/s. Delta answers "how much did it move the needle?"

4. Momentum, $p$ — "throwing power"

In the figure, momentum is drawn as an arrow whose length is $m\times v$. Notice the two ways to get a long arrow: lots of mass (big block) or high speed (long velocity arrow). Momentum bundles both into one number.

5. Conservation of momentum — "the total never changes"

The figure shows the skateboarder-in-space idea. Before the throw, person + ball sit still: total $p=0$. After the throw, the ball flies left with some negative momentum, so the person must fly right with equal positive momentum — the two arrows cancel to keep the total at $0$. Nobody pushed on the outside; they only pushed on each other.

6. Newton's law in momentum form — "push changes momentum"

7. The relative exhaust velocity, $v_e$

Two frames are drawn. Top: riding on the rocket, you always see gas leave your tail at the same speed $v_e$ — no matter how fast the whole rocket is going. Bottom: from the ground, the gas's speed is the rocket's speed $v$ minus $v_e$, written $v-v_e$, because the exhaust points backward.

8. The infinitesimal $d$ and the derivative $\frac{d}{dt}$

Think of a movie made of frames. $dt$ is the gap between two neighbouring frames — almost zero. In that gap the mass changes by $dm$ and the velocity by $dv$.

9. The natural logarithm, $\ln$

10. Mass ratio, $R$

Prerequisite map

Read the map top-down: mass and velocity build momentum; momentum plus Newton's law give conservation; conservation plus the exhaust speed plus calculus produce the derivation; the derivation spits out a logarithm; the log plus the mass ratio give the final formula.

Equipment checklist

Test yourself — cover the right side and answer aloud before revealing.