What is the Rocket Equation?
The Tsiolkovsky rocket equation is the fundamental equation of rocket propulsion. It describes how a rocket's velocity changes based on the speed of its exhaust gases and the ratio of its initial mass (with fuel) to its final mass (without fuel).
The equation reveals a key insight: to achieve higher velocities, you either need more efficient engines (higher exhaust velocity) or you need to carry more fuel relative to your payload.
The Formula
[\Delta v = v_e \times \ln\left(\frac{m_i}{m_f}\right)]
Where:
- ฮv is the change in velocity (delta-v)
- v_e is the exhaust velocity
- m_i is the initial mass (rocket + fuel)
- m_f is the final mass (rocket without fuel)
- ln is the natural logarithm
Calculation Example
Consider a rocket with:
- Exhaust velocity: 3,000 m/s
- Initial mass: 50,000 kg
- Final mass: 10,000 kg
Plugging into the formula:
[\Delta v = 3000 \times \ln\left(\frac{50000}{10000}\right)]
[\Delta v = 3000 \times \ln(5) = 3000 \times 1.609 = 4827 \text{ m/s}]
The rocket can achieve approximately 4,827 m/s of velocity change.
Quick Reference
| Parameter | Description |
|---|---|
| Exhaust Velocity | Speed of gases escaping the rocket (m/s) |
| Initial Mass | Mass of the rocket with fuel (kg) |
| Final Mass | Mass of the rocket after fuel is burnt (kg) |
| Delta-v | Achievable change in velocity (m/s) |
Why It Matters
The rocket equation explains why space travel is so challenging. The exponential relationship means that small increases in delta-v require disproportionately large amounts of fuel. This is why rockets are mostly fuel by mass and why multi-stage rockets are used for reaching orbit.