# (Perfectly) Elastic Collision

• A (perfectly) elastic collision is one that retains 100% of its initial total KE.
• As a result, the two bodies always separate from each other at the same speed as they approached each other. (see proof here)
• The outcome of an elastic collision can therefore be calculated through the equations:

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

• and

$\begin{array}{l}\text{RSoA}=\text{RSoS}\\{{u}_{1}}-{{u}_{2}}={{v}_{2}}-{{v}_{1}}\end{array}$

• Depicted below are the momentum and KE variations during a head-on (perfectly) elastic 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.
• The total KE drops to zero at one point during the collision, but returns to 100% by the end of the collision.
• The two masses approach and separate from each other at relative speed of 2u.

• Depicted below are the momentum and KE variations during a head-on (perfectly) elastic 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.
• The total KE drops to 50% at one point during the collision, but returns to 100% by the end of the collision.
• The two masses approach and separate from each other at relative speed of u.