AI Tutor
When a lump of clay drops onto the floor, it hits with a dull thud and stops dead in its tracks. Where did all of that incoming motion energy go?
Every collision in an isolated system strictly conserves total linear momentum, but kinetic energyβ€”the macroscopic energy of motion given by K=12mv2K = \frac{1}{2}mv^2β€”is almost never fully conserved.

The Fate of Lost Kinetic Energy

In an inelastic collision, some fraction of the system's kinetic energy converts into other internal energy forms during the impact. We quantify this energy dissipation as Ξ”K=Kfβˆ’Ki\Delta K = K_f - K_i.
The dissipated energy βˆ£Ξ”K∣|\Delta K| transforms directly into thermal energy QQ (atomic vibrations) and plastic deformation work WdefW_{\text{def}} (breaking bonds and permanently reshaping the material).
How do we find the absolute maximum kinetic energy a collision can possibly destroy? Let's force the colliding objects to stick together.

Perfectly Inelastic Collisions

A collision is perfectly inelastic when the two colliding bodies latch together and move with a single common final velocity vfv_f.
By conservation of momentum, m1v1+m2v2=(m1+m2)vfm_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f, giving the final velocity: vf=m1v1+m2v2m1+m2v_f = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}
Notice the term ΞΌ=m1m2m1+m2\mu = \frac{m_1 m_2}{m_1 + m_2}, known as the reduced mass, which simplifies two-body interactions into an equivalent single effective body.
The negative sign guarantees that kinetic energy is always lost (Ξ”K<0)(\Delta K < 0) whenever relative velocity exists between the bodies before impact.