Irodov Problem 1.201

May 19, 2026

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Problem Statement

A body of mass m₁ moving with velocity v₁ collides head-on with a stationary body of mass m₂. Find the velocity of each body after a perfectly elastic collision.

Given

m₁, v₁ (initial), m₂ at rest. Elastic collision.

Concepts & Formulas

Conservation of momentum: m₁v₁ = m₁v₁’ + m₂v₂’. Conservation of KE: ½m₁v₁² = ½m₁v₁’² + ½m₂v₂’². Elastic collision formulas apply.

Step-by-Step Solution

Step 1: v₁’ = v₁(m₁−m₂)/(m₁+m₂).
Step 2: v₂’ = 2m₁v₁/(m₁+m₂).
Step 3: Check momentum: m₁v₁(m₁−m₂)/(m₁+m₂) + m₂·2m₁v₁/(m₁+m₂) = m₁v₁[(m₁−m₂+2m₂)/(m₁+m₂)] = m₁v₁ ✓.

Worked Calculation

v₁’ = v₁(m₁−m₂)/(m₁+m₂). v₂’ = 2m₁v₁/(m₁+m₂).

Boxed Answer

v₁' = v₁(m₁−m₂)/(m₁+m₂); v₂' = 2m₁v₁/(m₁+m₂)

Physical Interpretation

For equal masses (m₁=m₂), the first body stops and the second moves with v₁ — a classic billiard-ball result. For m₁≫m₂, the heavy body barely slows.