Irodov Problem 1.141 – Inelastic Collision and Energy Loss

Problem Statement

A body of mass m₁ moving with velocity v₁ collides and sticks to a stationary body of mass m₂. Find the kinetic energy lost.

Given Information

• m₁ at v₁, m₂ at rest
• Perfectly inelastic collision

Physical Concepts & Formulas

$$v’=\frac{m_1v_1}{m_1+m_2},\quad \Delta T = T_i – T_f$$

Step-by-Step Solution

Step 1: Final velocity v’ = m₁v₁/(m₁+m₂).

Step 2: T_i = ½m₁v₁², T_f = ½(m₁+m₂)v’².

Step 3: ΔT = T_i − T_f = ½m₁m₂v₁²/(m₁+m₂) = T̃ (CM-frame KE).

Worked Calculation

ΔT = m₁m₂v₁²/[2(m₁+m₂)]

Answer

$$\boxed{\Delta T=\frac{m_1m_2v_1^2}{2(m_1+m_2)}}$$

Physical Interpretation

The lost KE equals the CM-frame kinetic energy — converted to heat, deformation, sound. This is the maximum possible energy loss consistent with momentum conservation.