Irodov Problem 1.203

Problem Statement

A nucleus of mass M at rest decays into two fragments. Fragment 1 has mass m₁ and kinetic energy T₁. Find the kinetic energy of fragment 2.

Given

M at rest. Decay → m₁ (speed v₁) + m₂ (speed v₂). m₂ = M−m₁. By momentum conservation: m₁v₁ = m₂v₂.

Concepts & Formulas

Momentum: m₁v₁ = m₂v₂. KE ratio: T₂/T₁ = m₂v₂²/(m₁v₁²) · (m₁/m₂) … T₁ = ½m₁v₁², T₂ = ½m₂v₂². From m₁v₁ = m₂v₂: p₁ = p₂ = p. T = p²/(2m). So T₁/T₂ = m₂/m₁.

Step-by-Step Solution

Step 1: p₁ = p₂ (momentum conservation, nucleus at rest).
Step 2: T = p²/(2m) → T₁ = p²/(2m₁), T₂ = p²/(2m₂).
Step 3: T₂ = T₁·m₁/m₂ = T₁·m₁/(M−m₁).

Worked Calculation

T₂ = T₁·m₁/(M−m₁).

Boxed Answer

T₂ = T₁·m₁/(M − m₁)

Physical Interpretation

In any two-body decay from rest, both fragments carry equal and opposite momenta, so the lighter fragment always has more kinetic energy.