Irodov Problem 1.206

Problem Statement

A rocket expels gases at velocity u relative to itself. Find the rocket’s speed after it has expelled a fraction η of its initial mass m₀.

Given

m₀ initial mass, u exhaust velocity (relative), η = Δm/m₀ expelled.

Concepts & Formulas

Tsiolkovsky equation: v = u·ln(m₀/m_f). m_f = m₀(1−η). v = u·ln(1/(1−η)).

Step-by-Step Solution

Step 1: Tsiolkovsky: v = u ln(m₀/m_f).
Step 2: Mass expelled = ηm₀, so m_f = m₀ − ηm₀ = m₀(1−η).
Step 3: v = u ln(m₀/(m₀(1−η))) = u ln(1/(1−η)) = −u ln(1−η).

Worked Calculation

v = u ln(1/(1−η)) = −u ln(1−η).

Boxed Answer

v = −u ln(1 − η)

Physical Interpretation

For η = 1−1/e ≈ 0.632, the rocket reaches speed u. To reach 2u, you need η = 1−e⁻² ≈ 0.865 — diminishing returns due to the logarithm.