Is it possible to have an inelastic collision in which all the kinetic energy of the colliding objects is lost? If so, give an example. If not, why not?

In classical physics, the principle of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act upon it. This law is crucial during collisions, where the momentum before and after the event should remain the same, even if the objects involved change velocity.

For inelastic collisions, the objects involved might stick together or be deformed, but the combined momentum of the system will be the same before and after the collision. To compute this, one can use the formula for total momentum, \( P_{total} = m_1v_1 + m_2v_2 \), where \( m \) represents mass and \( v \) represents velocity of the objects indexed by 1 and 2 before collision, and the total post-collision momentum must be equal to this value.

When discussing a collision, 'loss' of kinetic energy refers to its transformation from the energy of motion into other forms, and not to a disappearance of energy, which would violate the law of conservation of energy.

In a perfectly inelastic collision, the maximum amount of kinetic energy is converted. This conversion manifests as heat, sound, or potential energy if there's a change in shape (deformation energy). The calculation of kinetic energy is based on the formula \( KE = \frac{1}{2} mv^2 \), with \( m \) being the mass and \( v \) the velocity. Although kinetic energy is not conserved in an inelastic collision, the amount before and after the collision can be compared to determine how much energy has been transformed.

A completely inelastic collision is a specific type of inelastic collision where the colliding objects coalesce into a single mass post-collision. Contrary to elastic collisions where objects bounce off each other, a completely inelastic collision results in the maximum kinetic energy loss as no rebound occurs.

These collisions are modeled by using the conservation of momentum to predict the combined object's velocity post-collision. Since the objects stick together, they will move with a common velocity that can be calculated using the formula \( v_{final} = \frac{m_1v_1 + m_2v_2}{m_1 + m_2} \), where subscript notation has the same meaning as previously mentioned. This situation is perfectly illustrated when a dart sticks to a dartboard, hence losing all its initial kinetic energy by adhering and not bouncing back.

Energy transformation during an inelastic collision refers to the process where kinetic energy is not retained in the same form but is instead converted into other energy types.

In the context of collisions, energy may be converted into thermal energy due to friction or deformation, sound from the impact, or potentially into chemical energy if a reaction occurs between the colliding objects. It's vital to understand that while the form of the energy changes, the total energy within a closed system remains conserved according to the First Law of Thermodynamics. This energy transformation is an essential concept in understanding real-world scenarios such as car crashes or sports dynamics, where inelastic collisions frequently occur.