Irodov Problem 1.147 – Rocket with Gravity

Problem Statement

A rocket launches vertically from Earth surface with exhaust speed u and mass ejection rate β = dm/dt. Find velocity as a function of mass.

Given Information

• Exhaust speed u
• Mass ejection rate β
• Gravity g

Physical Concepts & Formulas

$$m\frac{dv}{dt}=\beta u – mg,\quad \frac{dm}{dt}=-\beta$$

Step-by-Step Solution

Step 1: With dm/dt = −β: m dv = −u dm − mg dt.

Step 2: t = (m₀−m)/β, so mg dt = g(m₀−m)dm/(−βm)·(−β)… integrating.

Step 3: v = u ln(m₀/m) − g(m₀−m)/β.

Worked Calculation

v = u ln(m₀/m) − g(m₀−m)/β

Answer

$$\boxed{v=u\ln\frac{m_0}{m}-\frac{g(m_0-m)}{\beta}}$$

Physical Interpretation

Gravity subtracts from the Tsiolkovsky velocity. Fast burn (large β) minimizes gravity losses. This is why rocket engines are run at full throttle during launch.