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A 5.00 g bullet is shot through a 1.00 kg wood block...

A $5.00 \mathrm{~g}$ bullet is shot through a $1.00 \mathrm{~kg}$ wood block suspended on a string $2.00 \mathrm{~m}$ long. The center of mass of the block rises a distance of $0.38 \mathrm{~cm}$. Find the speed of the bullet as it emerges from the block, along a horizontal, straight line, if its initial speed is $450 \mathrm{~m} / \mathrm{s}$.

Key Concepts

Inelastic Collisions

An inelastic collision is one in which some kinetic energy is transformed into other forms of energy (like heat or deformation), even though momentum is conserved. In the context of the bullet passing through the block, the interaction is partially inelastic since energy is lost or transferred during the collision, leading the bullet to emerge at a lower speed.

Pendulum Dynamics

Pendulum dynamics involve the conversion between kinetic and potential energy as the pendulum swings. When the block, acting as a pendulum bob, rises after being struck by the bullet, its kinetic energy is converted into gravitational potential energy. The maximum rise of the pendulum gives a measure of the initial kinetic energy imparted, which is essential for analyzing the collision.

Conservation of Momentum

In collision problems, particularly those where multiple objects interact over a short interval, the total linear momentum of the system is conserved. This concept applies even when kinetic energy is not entirely conserved, such as in inelastic collisions. In the described scenario, the bullet’s momentum before collision and its momentum along with that imparted to the block after collision must add up to the original total momentum of the system.

Mechanical Energy Conservation

When an object moves under the influence of gravity without non-conservative forces doing significant work, the system’s mechanical energy (the sum of its kinetic and potential energies) is conserved. In problems involving pendulums, like the rise of a suspended block, the kinetic energy transferred to the block is eventually converted into gravitational potential energy at the peak of its swing.

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