Two spheres of radius r move horizontally in opposite directions. The first has mass 2m and speed 2u, the second has mass m and speed 4u. The coefficient of restitution is (1/√3). The centres of the two spheres lie on two parallel lines, a distance r apart.
(i) show that at the moment of impact, the line of centres of the spheres make an angle of 30° with their previous lines of motion.
(ii) find the speeds of the spheres after impact.
conservation of momentum and coefficient of restitution
The Attempt at a Solution
the i axis is along the line of their centres at impact.
the j axis is vertical to the i axis.
∅ is angle with which the line of the centres of the spheres makes an angle with there previous lines of motion.(they are parallel)
there is no change in j.
conservation of momentum along the i axis.
2m(2ucos∅ ) + m(-4ucos∅) = 2m(v1) + m(v2)
0 = 2v1 + v2
v2 = -2v1
coefficient of restitution
( v1 -(-2v1) )/(2ucos∅ -( -4ucos∅ ) ) = 3v1/6ucos∅ = - 1/√3
v1 = (-2/√3)ucos∅
Im not exactly sure what to do from here I know that cosinverse(√3/2) = 30
so presumably I should be able to find v1 = u and cancel them.
Im guessing it has something to do with their centers being a distance r apart but i'm not exactly sure what to do with that information.
Any help would be appreciated.