A car moving at speed \(v\) undergoes a one-dimensional collision with an identical car initially at rest. The collision is neither elastic nor fully inelastic; \(5 / 18\) of the initial kinetic energy is lost. Find the velocities of the two cars after the collision.

When we encounter collision scenarios in physics, the principle of the conservation of momentum is a fundamental tool for analyzing the motion of objects before and after the impact. The law asserts that the total momentum of a closed system—meaning no external forces are acting on it—remains constant. This means that in our car collision problem, the total momentum before the impact equals the total momentum after the impact.

Mathematically, this is expressed as:
\( m_1v_1 + m_2v_2 = m_1v_{1f} + m_2v_{2f} \), where \( m_1 \) and \( m_2 \) are the masses of the two cars, \( v_1 \) and \( v_2 \) are the initial velocities, and \( v_{1f} \) and \( v_{2f} \) are the final velocities post-collision. In our exercise, the cars have identical masses, simplifying the equation to \( v = v_{1f} + v_{2f} \) after considering one car initially at rest and the other moving at velocity \( v \).

While momentum is always conserved in collisions, kinetic energy isn't necessarily kept the same unless the collision is perfectly elastic. Kinetic energy, which is the energy of motion, can be transformed into other forms of energy upon impact, such as sound or heat. The concept of conservation of kinetic energy is used when the collision is elastic, meaning no energy is lost to these non-mechanical forms.

In our textbook problem, the cars undergo a collision where some of the kinetic energy is indeed lost. If we begin with kinetic energy \( K_1 \) and lose some fraction to reach \( K_2 \), the conservation statement can be written as:
\( K_1 - \text{Energy lost} = K_2 \). Specifically, in our exercise, we have that \( \frac{5}{18} \) of the initial kinetic energy is lost, leaving \( \frac{13}{18} \) of the initial kinetic energy. This relationship helps us find the final velocities of the cars by examining the change in kinetic energy.

An inelastic collision is a type of collision in which the colliding objects do not conserve kinetic energy, though they do conserve momentum. This kind of collision results in some degree of deformation and/or generation of heat. In extreme cases, this could mean the colliding objects stick together and move with a common velocity after impact; these are perfectly inelastic collisions.

In our exercise, the collision is described as 'neither elastic nor fully inelastic,' indicating that it's a partially inelastic collision. During such an inelastic event, while the objects don't stick together, not all of the system's kinetic energy is conserved. This is vital to solve for the final velocities, as it affects the conservation of kinetic energy equation and tells us that only a portion of the initial kinetic energy remains post-collision.

During a collision, energy loss might seem puzzling since the law of conservation of energy says that energy cannot be created or destroyed. However, it's crucial to understand that it can be transformed from one form to another. In the context of collisions, while the total energy of the system is conserved, some of the kinetic energy—the energy associated with the objects' motions—is often converted into other forms due to forces during the collision.

Common energy transformations include kinetic energy turning into thermal energy (heat), sound energy, or potential energy in the form of deformation. Our problem specifies that \( \frac{5}{18} \) of the initial kinetic energy is lost, hinting at such transformations. Bearing this in mind when solving collision problems offers a more complete understanding of the dynamics involved and the actual outcomes for the motion of the objects after the collision.