The momentum of a closed system is a constant (as the law of conservation of momentum states). The full momentum of a system is the sum of momentums of all systems' bodies. Note that momentum is a vector quantity, mass multiplied by its velocity, and it is conserved as a vector, including direction.
Our system consists of a ring and two beads. Consider an axis along the initial speeds of beads (they are parallel), and projection...
The momentum of a closed system is a constant (as the law of conservation of momentum states). The full momentum of a system is the sum of momentums of all systems' bodies. Note that momentum is a vector quantity, mass multiplied by its velocity, and it is conserved as a vector, including direction.
Our system consists of a ring and two beads. Consider an axis along the initial speeds of beads (they are parallel), and projection of velocities on this axis. Momentum is a constant vector, so the projection of momentum on any axis is also a constant. Also it is evident from the symmetry of the situation that a ring will move only along the same axis. Denote the speed of a ring just before the collision as
The starting momentum's projection is (a ring is in rest). Just before the collision the beads' velocities are perpendicular to the chosen axis, so their projections are zero. Thus the momentum just before the collision is
This way we obtained a simple equation, or
This is the answer.
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