Irodov Problem 1.204

May 19, 2026

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Problem Statement

A particle of mass m₁ elastically collides with a stationary particle of mass m₂. Find the maximum angle of deflection of the first particle.

Given

m₁ > m₂. Elastic collision. Maximum scattering angle θ_max of m₁.

Concepts & Formulas

In CM frame, m₁ scatters at any angle. Transform to lab. Maximum deflection angle: sinθ_max = m₂/m₁ (when m₁ > m₂).

Step-by-Step Solution

Step 1: In CM frame, both particles scatter symmetrically.
Step 2: Lab angle of m₁: tanθ = (m₂ sinφ)/(m₁ + m₂ cosφ) where φ is CM angle.
Step 3: Maximize dθ/dφ = 0. For m₁ > m₂: θ_max = arcsin(m₂/m₁).
Step 4: For m₁ ≤ m₂: θ_max = π/2 (or 180° — full backscatter possible).

Worked Calculation

θ_max = arcsin(m₂/m₁) for m₁ > m₂.

Boxed Answer

θ_max = arcsin(m₂/m₁) for m₁ > m₂

Physical Interpretation

A heavy particle cannot be deflected by more than arcsin(m₂/m₁) in an elastic collision with a lighter target — it’s geometrically constrained by the momentum triangle.