Perfectly Inelastic Collisions

A perfectly inelastic collision is a specific type of collision where two objects impact and stick together rather than bouncing apart, moving as a single unit immediately after the event. The defining characteristic is that the relative velocity between the objects becomes zero, and they share a common final velocity vector $\vec{v}_f$.

Provided the system is isolated, linear momentum is conserved. If two masses $m_A$ and $m_B$ collide with initial velocities $\vec{v}_A$ and $\vec{v}_B$, the conservation of momentum relates the initial state to the final combined state:

$$m_A \vec{v}_A + m_B \vec{v}_B = (m_A + m_B)\vec{v}_f$$

While momentum is conserved, kinetic energy is not. A perfectly inelastic collision results in the maximum possible loss of kinetic energy that still conserves momentum. The initial kinetic energy $K_i$ is the sum of the individual energies:

$$K_i = \frac{1}{2}m_A v_A^2 + \frac{1}{2}m_B v_B^2$$

After the collision, the final kinetic energy $K_f$ involves the combined mass and the new common speed:

$$K_f = \frac{1}{2}(m_A + m_B) v_f^2$$

The lost energy, calculated as $K_i - K_f$, is transformed into other forms such as thermal energy or work done permanently deforming the materials.

In physics problems, specific language often signals this interaction. Look for keywords such as "sticks to," "embeds," "couples," or "latches onto." These terms indicate that the objects join to form a single system with mass $m_\mathrm{total} = m_A + m_B$.

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