Problem Statement
Find the speed of a satellite at any point of its elliptical orbit with semi-major axis a, when it is at distance r from Earth’s center.
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. Vis-viva equation: $v^{2}$ = $GM_E(2/r) – 1/a$).
- $F = G\dfrac{m_1 m_2}{r^2}$ — 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.
$$Total energy of ellipse: E = -GM_Em/(2a)$$
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.
$$E = (1/2)mv^{2}- GM_Em/r$$
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.
$$(1/2)v^{2}= GM_E(1/r – 1/(2a)) → v^{2}= GM_E(2/r – 1/a)$$
Worked Calculation
$$v = √[GM_E(2/r – 1/a)]$$
Answer
$$\boxed{v = √[GM_E(2/r – 1/a)] (vis-viva equation)}$$
This is the quantity the problem asked for, expressed in terms of the given data: $v = √[GM_E(2/r – 1/a)] (vis-viva equation)$.
Physical Interpretation
The vis-viva equation is the single most useful formula in orbital mechanics — it gives speed anywhere in the orbit from just the current distance and orbit size. 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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