Use the result in Problem 57 to find the muzzle velocity
Reference Problem 57
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 mw is attached. The system is set in motion by the impact of a bullet that is moving horizontally at the unknown muzzle velocity vb; at the time of the impact, t = 0, the combined mass is mw + mb, where mb 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 vb can be found by measuring the height h of the mass mw + mb at the maximum displacement angle θmax shown in FIGURE 3.R.3.
Intuitively, the horizontal velocity V of the combined mass mw + mb after impact is only a fraction of the velocity vb of the bullet, that is,
(a) Solve the initial-value problem
(b) Use the result from part (a) to show that
(c) Use Figure 3.R.3 to express cos θmax in terms of l and h. Then use the first two terms of the Maclaurin series for cos θ to express θmax in terms of l and h. Finally, show that vb is given (approximately) by
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