Problem Statement
A comet moves in a very elongated ellipse. At perihelion it is at distance $r_p$ from the Sun (mass M) with speed $v_p$. Find its speed at aphelion $r_a$.
Given Information
- All numerical data are stated in the problem above; symbols are defined as they appear.
Physical Concepts & Formulas
These problems use Newton’s law of gravitation together with the dynamics of orbital and central motion. Also energy conservation.
- $F = G\dfrac{m_1 m_2}{r^2}$ — law of universal gravitation
- $v = \sqrt{\dfrac{GM}{r}}$ — circular orbital speed
- $T^2 = \dfrac{4\pi^2 r^3}{GM}$ — Kepler’s third law
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.
$$L = m \cdot v_p \cdot r_p = m \cdot v_a \cdot r_a → v_a = v_p \cdot r_p/r_a$$
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. Substitute the quantities listed under Given Information into that relation, keeping symbols algebraic for now
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. Solve the resulting equation for the unknown the problem asks for and simplify to a clean closed form
Worked Calculation
$$v_a = v_p \cdot r_p/r_a$$
Answer
$$\boxed{v_a = v_p \cdot r_p/r_a}$$
This is the quantity the problem asked for, expressed in terms of the given data: $v_a = v_p \cdot r_p/r_a$.
Physical Interpretation
Conservation of angular momentum for a central force gives this elegant result: speed is inversely proportional to distance at orbital extremes. Kepler’$s 2nd$ law in action. 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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