Irodov Problem 1.145 – Rocket Motion and Tsiolkovsky Equation

Problem Statement

A rocket ejects gas at speed u relative to itself. Starting from rest, find the speed after ejecting mass Δm (initial mass m₀).

Given Information

• Exhaust speed: u relative to rocket
• Initial mass m₀
• Final mass m₀ − Δm

Physical Concepts & Formulas

$$v=u\ln\frac{m_0}{m}$$

Step-by-Step Solution

Step 1: Tsiolkovsky equation: v = u ln(m₀/m_f).

Step 2: m_f = m₀ − Δm.

Step 3: v = u ln(m₀/(m₀−Δm)) ≈ u(Δm/m₀) for small Δm.

Worked Calculation

v = u ln[m₀/(m₀−Δm)]

Answer

$$\boxed{v=u\ln\frac{m_0}{m_0-\Delta m}}$$

Physical Interpretation

The rocket equation shows that speed gain requires exponential fuel mass. This is why chemical rockets must carry so much propellant for orbital missions.