We consider the collision between a flat piece (mass M) and a sphere (mass m). At the first moment of contact the speed of the flat piece is V and the sphere is at rest. At the last moment of interaction, the speed of the sphere is x and V become V'. We suppose that all these speeds point in the same direction. The linear momentum conservation gives
MV=MV'+mx (1)
Be e=Ec'/Ec the coefficient of restitution in term of kinetic energy. We have
0.5eMV^2 = 0.5MV'^2+0.5mx^2 (2)
By eliminating V' between (1) and (2) we get to
(m^2+Mm)x^2-2MmVx=(e-1)(MV)^2 (3)
So we have 2 solutions for x if
e>1-m/(m+M)
and no solution else.
What does this condition means physically ? And why is there 2 solutions a priori ?
Thanks a lot.