Irodov Problem 1.155 – Areal Velocity and Kepler’s Second Law

Problem Statement

A particle moves under a central force with angular momentum L and mass m. Find the areal velocity dA/dt.

Given Information

• Angular momentum L = const
• Mass m
• Central force

Physical Concepts & Formulas

$$\frac{dA}{dt}=\frac{L}{2m}$$

Step-by-Step Solution

Step 1: Area swept in dt: dA = ½|r × dr| = ½|r × v|dt.

Step 2: |r × v| = L/m.

Step 3: dA/dt = L/(2m).

Worked Calculation

dA/dt = L/(2m)

Answer

$$\boxed{\frac{dA}{dt}=\frac{L}{2m}}$$

Physical Interpretation

This is Kepler’s second law: equal areas in equal times. It follows purely from angular momentum conservation, valid for any central force, not just gravity.