# Perfectly Inelastic Collision

• A (perfectly) inelastic collision is one that retains the minimum amount of its initial total KE.
• As a result, the two bodies will always “stick together” and travel at a common speed after the collision.
• The outcome of an elastic collision can therefore be calculated using the PCOM equation:

$\displaystyle \begin{array}{c}\sum {{p}_{i}}=\sum {{p}_{f}}\\{{m}_{1}}{{u}_{1}}+{{m}_{2}}{{u}_{2}}=({{m}_{1}}+{{m}_{2}})v\end{array}$

• Depicted below are the momentum and KE variations during a head-on perfectly inelastic collisions of two equal masses m with equal initial speed u.

• For simplicity, we assume a constant contact force during the collision.
• Note that the total momentum remains constant (at zero) throughout the collision.
• To conserve momentum, both masses must come to rest after the collision, losing 100% of the initial KE.

• Depicted below are the momentum and KE variations during a head-on perfectly inelastic collision of two equal masses m, one of them with initial speed of u, and the other initially at rest.

• For simplicity, we assume a constant contact force during the collision.
• Note that the total momentum remains constant (at mu) throughout the collision.
• To conserve momentum, both masses must travel at a common speed of 0.5u after the collision, losing 50% of the initial total KE.
• (Note that it is not possible to lose more than 50% because of momentum conservation)