Direct impact of two smooth spheres:
A smooth sphere of mass m1 impinges directly with velocity u1 on another smooth sphere of mass m2, moving in the same direction with velocity u2; if the coefficient of restitution is e, to find their velocities after the impact:
AB is the line of impact, i.e. the common normal. Due to the impact there is no tangential force and hence, for either sphere the velocity along the tangent is not altered by impact. But before impact, the spheres had been moving only along the line AB (as this is a case of direct impact).
Hence for either sphere tangential velocity after impact = its tangent velocity before impact = 0. So, after impact, the spheres will move only in the direction AB. Let their velocities be v1 and v2.
By Newton's experimental law, the relative velocity of m2 with respect to m1 after impact is (-e) times the corresponding relative velocity before impact.
Note;
If one sphere say m2 is moving originally is a direction opposite to that of m1, the sign of u2 will be negative. Also it is most important that the directions of v1 and v2 must be specified clearly.
Usually we take the positive direction as from left to right and then assume that both v1 and v2 are in this direction. If either of them is actually in the opposite direction, the value obtained for it will turn to be negative.
In writing equation (1) corresponding to Newton's law, the velocities must be subtracted in the same order on both sides. In all problems it is better to draw a diagram showing clearly the positive direction and the directions of the velocities of the bodies.
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