A fascinating case is that of two balls connected (or separated) by a spring that collide. The collision happens in three clearly distinct phases, and following them one by one is the best way to understand what really happens inside an elastic collision.
The three phases: approach (A), maximum compression (B, both at ), separation (C, spring at rest again).
The key is state B, maximum compression. At that instant the two bodies have the same velocity: that of the centre of mass . If they had different velocities the distance between them would still be changing, and the spring would not be at maximum compression. In the centre-of-mass frame, then, both are momentarily at rest: all the internal kinetic energy (that of the relative motion) has turned into elastic potential energy of the spring. It is from this equality — initial internal energy equal to spring energy — that the maximum compression is found.
When the spring re-expands and returns to its rest length (state C), it gives back all the energy: from A to C the overall process is an elastic collision, and the trick (reversal of the relative velocity) holds.
The full calculation with numbers is in Esercizio svolto — collisione tramite molla.
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Topics: Quantità di moto e urti Concepts: Urto elastico · Conservazione della quantità di moto · Energia cinetica · Energia potenziale elastica Skills: Metodo delle due tabelle Objects: Molla
Related exercises: Esercizio svolto — collisione tramite molla · Urto elastico tra due biglie · Dimostrazione urto elastico 1D