Irodov Problem 1.146 – Rocket Velocity and Fuel Mass Ratio

Problem Statement

A rocket in free space (no gravity) ejects gas at speed u = 3.0 km/s relative to rocket. Find the mass ratio m₀/m_f needed to reach v = 12 km/s.

Given Information

• Exhaust velocity u = 3.0 km/s
• Final velocity v = 12 km/s

Physical Concepts & Formulas

$$v=u\ln(m_0/m_f)\Rightarrow m_0/m_f=e^{v/u}$$

Step-by-Step Solution

Step 1: v/u = 12/3 = 4.

Step 2: m₀/m_f = e⁴ ≈ 54.6.

Step 3: About 98% of initial mass must be propellant.

Worked Calculation

m₀/m_f = e^4 ≈ 54.6

Answer

$$\boxed{m_0/m_f=e^{v/u}=e^4\approx 54.6}$$

Physical Interpretation

To reach 4 times exhaust velocity, you need 54× mass ratio. This exponential dependence makes high-Δv missions extremely fuel-intensive.