Tsiolkovsky Rocket Equation

1. Overview

The Tsiolkovsky Rocket Equation describes how a rocket’s velocity changes as it expels mass in the form of propellant. Under idealized assumptions—no external forces (for example, no gravity or drag) and a constant exhaust velocity—it relates the rocket’s change in velocity to its initial and final masses and the effective exhaust velocity.

A common final form of the Rocket Equation is:

$$ \Delta v = v_e \,\ln\!\Bigl(\frac{m_0}{m_f}\Bigr) $$

where

• $m_0$ is the initial mass of the rocket (including all propellant),
• $m_f$ is the final mass of the rocket (after some or all propellant is used up),
• $v_e$ is the effective exhaust velocity (the relative speed at which propellant leaves the rocket),
• $\Delta v$ is the net change in the rocket’s velocity under these ideal conditions.

2. Fundamental Assumptions

1. No external forces
   The rocket is assumed to be in free space with negligible gravity and drag.
2. Constant exhaust velocity
   The propellant is expelled at a constant speed $v_e$ relative to the rocket.
3. Instantaneous mass flow
   During a brief interval $\mathrm{d}t$, the rocket’s mass changes by a small $\mathrm{d}m$ (negative, since the rocket is losing mass). The change in momentum is thus governed by the expelled propellant alone.

Under these assumptions, momentum conservation applies in each short interval of propellant ejection.

3. Momentum Conservation (Differential Form)

Consider a small time interval $\mathrm{d}t$:

• Before ejection:
• Mass: $m$
• Velocity: $v$
• Momentum: $m \, v$
• After ejection:
• Mass: $m - \mathrm{d}m$
• Velocity: $v + \mathrm{d}v$
• Expelled propellant mass: $\mathrm{d}m$
• Velocity of expelled propellant (inertial frame): $v - v_e$

Hence, the final momentum is

$$ (m - \mathrm{d}m)\,(v + \mathrm{d}v) \;+\; \mathrm{d}m \,\bigl(v - v_e\bigr). $$

Because no external force acts on the system (idealized), momentum is conserved:

$$ m\,v \;=\; (m - \mathrm{d}m)\,(v + \mathrm{d}v) \;+\; \mathrm{d}m \,\bigl(v - v_e\bigr). $$

4. Simplifying and Isolating $\mathrm{d}v$

Expand and neglect the second-order small term $\mathrm{d}m \,\mathrm{d}v$:

$$ (m - \mathrm{d}m)\,(v + \mathrm{d}v) \;\approx\; m\,v + m\,\mathrm{d}v - \mathrm{d}m\,v. $$

Then the right-hand side becomes

$$ m\,v + m\,\mathrm{d}v - \mathrm{d}m\,v + \mathrm{d}m\,\bigl(v - v_e\bigr) \;=\; m\,v + m\,\mathrm{d}v - \mathrm{d}m\,v_e. $$

Equating to the left side $m\,v$:

$$ m\,v \;=\; m\,v + m\,\mathrm{d}v - \mathrm{d}m\,v_e. $$

Subtract $m\,v$ from both sides:

$$ 0 \;=\; m\,\mathrm{d}v - \mathrm{d}m\,v_e, $$

which implies

$$ m\,\mathrm{d}v = v_e \,\mathrm{d}m. $$

5. Separating Variables and Integrating

Rearrange:

$$ \mathrm{d}v = v_e \,\frac{\mathrm{d}m}{m}. $$

Integrate from $m_0$ down to $m_f$ (rocket mass before and after propellant burn), and from velocity $v_i$ to $v_f$:

$$ \int_{v_i}^{v_f} \mathrm{d}v \;=\; v_e \int_{m_0}^{m_f} \frac{\mathrm{d}m}{m}. $$

Left side:

$$ v_f - v_i. $$

Right side:

$$ v_e \,\ln\!\Bigl(\frac{m_f}{m_0}\Bigr). $$

Hence,

$$ v_f - v_i = v_e \,\ln\!\Bigl(\frac{m_f}{m_0}\Bigr). $$

If the rocket starts from rest ($v_i = 0$), then

$$ v_f = v_e \,\ln\!\Bigl(\frac{m_0}{m_f}\Bigr). $$

Defining $\Delta v = v_f - v_i$ gives us the Tsiolkovsky Rocket Equation:

$$ \Delta v = v_e \,\ln\!\Bigl(\frac{m_0}{m_f}\Bigr). $$

6. Physical Interpretation

• A larger $\Delta v$ results from making the ratio $\tfrac{m_0}{m_f}$ bigger (i.e., carrying more propellant relative to final mass) or increasing $v_e$ (the engine’s exhaust velocity).
• In realistic scenarios with gravity or air resistance, additional terms appear to account for these effects. Nonetheless, the Tsiolkovsky equation remains the foundation of classical rocket mechanics.