Rocket and the Tsiolkovsky equation

Problem

Rocket and Tsiolkovsky. A rocket has initial velocity $0$ and final velocity $200\;\text{m/s}$ relative to the Earth. It ejects gas at $v_g = 2\;\text{km/s}$ relative to the rocket. Using $\Delta v = v_g\ln(m_0/m_f)$, calculate the ratio between the initial and final mass.

Solution

Applying the Tsiolkovsky equation with $\Delta v = 200\;\text{m/s}$ and $v_g = 2000\;\text{m/s}$: $\Delta v = v_g \ln\!\frac{m_0}{m_f} \;\Rightarrow\; \ln\!\frac{m_0}{m_f} = \frac{200}{2000} = 0{,}1$ $\frac{m_0}{m_f} = e^{0{,}1}$ $\ev{\frac{m_0}{m_f} \approx 1{,}105}$ About $10{,}5\%$ extra mass (as propellant) is needed to gain those $200\;\text{m/s}$.

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Topics: Momentum and collisions Concepts: Conservation of momentum · Explosion