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General Physics I:

Subsection 5.4.2 Ballistic Pendulum

Figure 5.4.3.
From the principle of conservation of momentum,
\begin{equation*} \text{Momentum before collision = Momentum after collision.} \end{equation*}
\begin{equation*} mv_{o}+0 = (M+m)v \end{equation*}
\begin{equation} \therefore v=\frac{mv_{o}}{M+m} \tag{5.4.19} \end{equation}
From the principle of conservation of energy,
\begin{equation*} E_{A} = E_{B}. \end{equation*}
\begin{equation*} \text{or,}\quad \frac{1}{2}(M+m)v^{2}+0=0+(M+m) gy \end{equation*}
\begin{equation} \therefore v=\sqrt{2gy} \tag{5.4.20} \end{equation}
From eqns. (5.4.19) and (5.4.20), we have -
\begin{equation*} \frac{mv_{o}}{M+m} = \sqrt{2gy} \end{equation*}
\begin{equation*} \therefore \quad v_{o} = \left(\frac{M+m}{m}\right)\sqrt{2gy} = \left(1+\frac{M}{m}\right)\sqrt{2gy} \end{equation*}
But
\begin{equation*} y=l-l\cos\theta = l(1-\cos\theta). \end{equation*}
\begin{equation} \therefore \quad v_{o}=\left(1+\frac{M}{m}\right)\sqrt{2gl(1-\cos\theta)} \tag{5.4.21} \end{equation}
Where \(v_{o}\) is called the muzzle velocity.