Two gliders on a linear air track, no friction. they been added with springs so collision will be elastic?

Question

initial velocity of m1 = 2m/s

mass of m1 = 0.5 kg

intial velocity of m2 = -2m/s

mass of m2 = 0.3kg

final velocity of m1 and m2 = ?

Answer 1

♣initial momentum L=m1*v1 +m2*v2;

final momentum L=m1*v3 +m2*v4;

and momentum conserves: L=L;

(1) 0.5*v3 +0.3*v4 =0.5*2 –0.3*2 =0.4 kg·m/s;

♠initial kinetic energy E=0.5*m1*v1^2 +0.5m2*v2^2;

final kinetic energy E=0.5*m1*v3^2 +0.5m2*v4^2;

and energy conserves: E=E;

(2) 0.5*v3^2 +0.3*v4^2 =0.5*2^2 +0.3*2^2 = 3.2 J;  

♦the task is to solve system of 2 eqtns (1)&(2);

from (1) we get v3=(4 –3*v4)/5;

eqn(2) now looks: 5*((4 –3*v4)/5)^2 +3*v4^2 = 32, hence

v4=(1±5)/2 = -2, +3; hence v3= 2, -1;

♥ plugging the pair v3=2, v4=-2 into (1) we get:

0.5*2 +0.3*(-2) = 0.4; correct! But

this is what was initially! So we discard this pair;  

♥ plugging the pair v3=-1, v4=3 into (1) we get:

0.5*(-1) +0.3*3 = 0.4; correct!  

This is what happened after bouncing;