Irodov Problem 1.142 – Ballistic Pendulum

Problem Statement

A bullet of mass m moving at v embeds in a pendulum of mass M suspended at length L. Find the angle of swing.

Given Information

• Bullet: mass m, velocity v
• Pendulum: mass M, length L

Physical Concepts & Formulas

$$mv=(m+M)v’,\quad (m+M)gL(1-\cos\theta)=\tfrac{1}{2}(m+M)v’^2$$

Step-by-Step Solution

Step 1: Momentum conservation: v’ = mv/(m+M).

Step 2: Energy conservation for swing: ½(m+M)v’² = (m+M)gL(1−cosθ).

Step 3: cosθ = 1 − v’²/(2gL) = 1 − m²v²/[2gL(m+M)²].

Worked Calculation

θ = arccos[1 − m²v²/(2gL(m+M)²)]

Answer

$$\boxed{\theta=\arccos\!\left[1-\frac{m^2v^2}{2gL(m+M)^2}\right]}$$

Physical Interpretation

The ballistic pendulum is a classic device for measuring bullet speed. Momentum is conserved during impact; energy is conserved during the swing.