A ballistic pendulum is a device that was used to measure the speeds of bullets before the development of electronic timing devices. The device consists of a large block of wood of mass \( M \), hanging from two long cords. A bullet of mass \( m \) is fired into the block, the bullet comes quickly into rest and the block + bullet rises to a vertical distance \( h \) before the pendulum comes momentarily to rest as the ends of the arc.
In the process, the linear momentum is conserved. In such a collision, some kinetic energy is dissipated as heat; so mechanical energy is not conserved. When there is a loss in mechanical energy, the collision is said to be inelastic. Further when two bodies coalesce, the collision is said to be perfectly inelastic.
What is the initial speed of the bullet in terms of height \( h \) ?
(1) \( \frac{m}{M} \sqrt{2 g h} \)
(2) \( \frac{m}{M+m} \sqrt{2 g h} \)
(3) \( \frac{M+m}{m} \sqrt{2 g h} \)
(4) \( \sqrt{\frac{2 m}{M+m} g h} \)
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