Collars A and B of the same mass m are moving toward each ot

Collars A and B, of the same mass m, are moving toward each other with the velocities shown. Knowing that the coefficient of restitution between the collars is 0 (plastic impact), show that after impact: The common velocity of the collars is equal to HALF the difference in their speeds before impact. The loss in kinetic energy is given by the equation: 1/4 m(v_A + v_B)^2

Solution

Mass m = m

Initial velocities u = v_A

                     U = - v_B

Here negative sign indicates that it moves opposite to the another.

Coefficient of restitution is zero.i.e., it is perfectly inelastic collision.

From law of conservation of linear momentum ,

mu +mU = (m+m) v

   u + U = 2v

v_A-v_B = 2v

From this combined velocity after collision v = (v_A-v_B) / 2

(b). Total initial kinetic energy K = (1/2) mv_A² + (1/2) mv_B²

Total final kinetic energy K \' = (1/2) (m+m)v ²

                                        = (1/2)(2m)[ (v_A-v_B) / 2] ²

Loss in kinetic energy = K - K \'

                                = (1/2) mv_A² + (1/2) mv_B² -(1/2)(2m)[ (v_A-v_B) / 2] ²

                                          = (1/2)m {[v_A² + v_B² -2[ (v_A²+v_B² -2v_Av_B) / 4]}

                               = (1/2)m{ [v_A² + v_B² -[ (v_A²+v_B² -2v_Av_B) / 2]}

                               = (1/2)m { [v_A² + v_B² - (1/2)v_A²-(1/2)v_B² +v_Av_B) }

                               = (1/2)m { [(1/2)v_A²+(1/2)v_B² +v_Av_B) }

                               = (1/4)m { [v_A²+v_B² +2v_Av_B]}

                               = (1/4) (v_A+v_B) ²