Perfectly Inelastic Collisions

Collisions where bodies stick together and kinetic-energy loss is maximal.

Conceptual context

A perfectly inelastic collision is the extreme case where bodies unite after impact and move as a single object. It is the collision that loses the maximum possible kinetic energy without violating momentum conservation. In this model, the coefficient of restitution is zero.

Although it seems like a "simple" case (one unknown), it is of enormous practical relevance: it models a projectile embedding itself in a block, railway cars coupling, a meteorite hitting a planet, and particle capture in nuclear physics.

🟢 Essential level

The essential idea is not the formula but the physical result: after impact, two independent motions no longer exist. The bodies form a single assembly and therefore share a single final velocity.

That common velocity does not mean the collision was "soft." It means the system gives up conserving separate motions. The total momentum still determines how the assembly moves forward, but part of the initial kinetic energy $K_i$ is no longer seen as translation and is transformed into deformation, heat, or sound.

It is important at this level to recognize two simultaneous facts. First, the assembly keeps moving because momentum does not disappear. Second, the collision no longer returns the bodies as separate objects, so the final motion is poorer than the initial one from an energetic point of view.

If a light body arrives very fast against another much heavier one, the final assembly usually moves slowly. If the masses are similar, the common velocity may be more intermediate and the energy loss is especially visible. This is the intuition that should be clear before moving on to mathematical expressions.

[!IMPORTANT] The final velocity is always between the initial velocities of the two bodies. It can never be greater than the largest nor less than the smallest of the initial velocities.

🔵 Formal level

Basic formulas

Energy balance and dissipation

The fundamental expression of the model is the linear momentum balance:

Conservation of linear momentum (Inelastic collision)

This balance determines the common final velocity $v_f$ of the assembly:

Common final velocity

From this final state, we calculate the energetic magnitudes of the leaf. The energy available before impact is $K_i$:

Initial kinetic energy

The kinetic energy that survives as translation of the joined assembly is $K_f$:

Final kinetic energy

And the amount of degraded mechanical energy is quantified by the variation $\Delta K$:

Change in kinetic energy

Since it is a perfectly inelastic collision, $\Delta K$ is always negative (or zero in the trivial case): the final kinetic energy $K_f$ is strictly less than the initial $K_i$.

To evaluate the severity of the loss, we usually calculate the fraction of energy that has been transformed into non-mechanical forms:

Dissipated energy fraction

If the target is at rest, this fraction depends exclusively on the mass ratio. If the target $m_2$ is much heavier than the projectile $m_1$, almost all the energy is dissipated. If the projectile is dominant, most of the energy is conserved as translation of the assembly.

In the leaf's vocabulary, $v_f$ sets the shared motion, $K_i$ sets the starting energy scale, $K_f$ identifies the useful part that survives, and $\Delta K$ along with $\xi$ characterize the dissipative nature of the impact.

Real student questions

• Is a perfectly inelastic collision always the one that loses the most energy? Yes, for a given mass and momentum. It is the lower limit for the final kinetic energy of the system.
• Can one body absorb another without deforming? In principle, yes (a sandbag catching a projectile), but there is always internal dissipation.
• Can the elastic formula be used with a zero coefficient of restitution? Yes: the general formulas with a coefficient of restitution give identical velocities for both bodies when that coefficient is zero, which is the formula for the perfectly inelastic collision.

Final synthesis

The perfectly inelastic collision (coefficient of restitution of zero) unites bodies into a single object, losing the maximum kinetic energy compatible with momentum conservation. Its resolution is algebraically the simplest (one equation, one unknown) and has direct applications in ballistics, impact engineering, and particle physics.