Cars B and C are at rest with their breaks off Car A plows i

Cars B and C are at rest with their breaks off. Car A plows into car B at high speed, and this causes car B to hit C. The collisions are completely inelastic (i.e. the cars stick together). What fraction of the initial energy (the energy present in the system before any collisions have happened) is dissipated in the second collision alone (when cars A and B which are already stuck together collide with car C)? The cars are of identical construction.

Solution

By Conservation of momentum in each collision

Initial Mechanical energy

E₁=P²/2m

Mechanical energy after 1st collision

E₂ =P²/2(2m) =P²/4m

Mechanical energy after 2nd collision

E₃ =P²/2(3m) =P²/6m

Fraction of initial energy lost in 2nd collision is

E₂-E₃/E₁ =(P²/4m)-(P²/6m)/(P²/2m)

E₂-E₃/E₁ = 1/6