Irodov Problem 1.215

Problem Statement

A planet moves in an elliptical orbit around the Sun. At perihelion it is at distance r₁ and has speed v₁. Find its speed v₂ at aphelion (distance r₂).

Given

Ellipse: r₁ = perihelion, r₂ = aphelion. v₁ at perihelion. Gravity is central force.

Concepts & Formulas

Angular momentum conservation (Kepler’s 2nd law): L = m·v₁·r₁ = m·v₂·r₂ (since v ⊥ r at both extrema).

Step-by-Step Solution

Step 1: At perihelion and aphelion, velocity is perpendicular to radius vector.
Step 2: L = mv₁r₁ = mv₂r₂.
Step 3: v₂ = v₁r₁/r₂.

Worked Calculation

v₂ = v₁r₁/r₂.

Boxed Answer

v₂ = v₁r₁/r₂

Physical Interpretation

The planet moves fastest at perihelion and slowest at aphelion — a direct consequence of angular momentum conservation and the geometry of the ellipse.