Historically, in order to maintain quality control over munitions (bullets) produced by...

Historically, in order to maintain quality control over munitions (bullets) produced by an assembly line, the manufacturer would use a ballistic pendulum to determine the muzzle velocity of a gun; that is, the speed of a bullet as it leaves the barrel. The ballistic pendulum (invented in 1742), is simply a plane pendulum consisting of a rod of negligible mass to which a block of wood of mass m_w is attached. The system is set in motion by the impact of a bullet that is moving horizontally at the unknown muzzle velocity v_b; at the time of the impact, t = 0, the combined mass is m_w + m_b, where m_b is the mass of the bullet embedded in the wood. We have seen in (7) of Section 3.10 that in the case of small oscillations, the angular displacement θ(t) of a plane pendulum shown in Figure 3.11.3 is given by the linear corresponds to motion to the right of vertical. The velocity v_b can be found by measuring the height h of the mass m_w + m_b at the maximum displacement angle θ_max shown in FIGURE 3.R.3.

Intuitively, the horizontal velocity V of the combined mass m_w + m_b after impact is only a fraction of the velocity v_b of the bullet, that is,