Problem Statement
A rocket expels gases at velocity u relative to itself. Find the rocket’s speed after it has expelled a fraction $\eta$ of its initial mass $m_0$.
Given Information
- All numerical data are stated in the problem above; symbols are defined as they appear.
Physical Concepts & Formulas
These problems are governed by the kinematic relations connecting displacement, velocity and acceleration. We choose a convenient reference frame and integrate (or differentiate) the motion equations. Tsiolkovsky equation: $v = u \cdot \ln (m_0/m_f)$).
- $\vec{v} = \dfrac{d\vec{r}}{dt}$ — instantaneous velocity
- $\vec{a} = \dfrac{d\vec{v}}{dt}$ — instantaneous acceleration
- $\vec{r} = \vec{r}_0 + \vec{v}_0 t + \tfrac{1}{2}\vec{a}t^2$ — uniformly accelerated motion
Step-by-Step Solution
Step 1 — Identify the governing principle: We begin by recognising which physical law controls the situation and why it is the correct starting point for this problem.
$$Tsiolkovsky: v = u \ln (m_0/m_f)$$
Step 2 — Set up the relevant equations: Next we write down the equations that follow from that principle, introducing the symbols we will carry through the algebra. Mass expelled = $\eta$ $m_0$, so $m_f$ = $m_0 – \eta m_0$ = $m_0(1- \eta)$ )
Step 3 — Apply the given conditions: We now substitute the specific conditions and constraints given in the problem so the equations describe this particular situation.
$$v = u \ln (m_0/(m_0(1- \eta ))) = u \ln (1/(1- \eta )) = -u \ln (1- \eta )$$
Worked Calculation
$$v = u \ln (1/(1- \eta )) = -u \ln (1- \eta )$$
Answer
$$\boxed{v = -u \ln (1 – \eta )}$$
This is the quantity the problem asked for, expressed in terms of the given data: $v = -u \ln (1 – \eta )$.
Physical Interpretation
For $\eta$ = $1-1/e$ ≈ $0.632$, the rocket reaches speed u. To reach $2u$, you need $\eta$ = 1-$e^{$-2}$ ≈ $0.865$ — diminishing returns due to the logarithm. The magnitude of the answer is consistent with everyday physical experience for this class of problem in Irodov’s Part 1 — the result shows how the answer scales with the given quantities. If we doubled the dominant input, the boxed formula tells us exactly how the output would respond, and that scaling is the key physical insight this problem trains. Comparing the answer with the appropriate limiting cases (very small or very large values of the dominant parameter) recovers the familiar Newtonian or intuitive expectation, which is a useful sanity check.
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