(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.

collisionGraphs A

  • 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.

collisionGraphs C

  • 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.

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