Solving 100% Elastic and Inelastic collision

{{dir1}}

{{dir2}}

100% Elastic First Approach - Solve for V2f first

{{m1}} * {{v1}} + {{m2}} * {{v2}} = {{m1}} * V1f + {{m2}} * V2f
V1f = ({{m1}} * {{v1}} + {{m2}} * {{v2}} - {{m2}} * V2f) / {{m1}}
V1f = ({{p}} - {{m2}} * V2f) / {{m1}} = {{pOverm1}} - {{m2Overm1}} * V2f
12 * m1 * V12 + 12 * m2 * V2 2 = 12 * m1 * V1f2 + 12 * m2 * V2f 2
12 * {{m1}} * {{v1}}2 + 12 * {{m2}} * {{v2}} 2 = 12 * {{m1}} * V1f2 + 12 * {{m2}} * V2f 2
{{ke1}} = 12 * {{m1}} * V1f2 + 12 * {{m2}} * V2f 2
{{ke1}} = {{m12}} * V1f2 + {{m22}} * V2f2
{{ke1}} = {{m12}} * ( {{pOverm1}} - {{m2Overm1}} * V2f))2 + {{m22}} * V2f 2
{{ke1}} = {{m12}} * ({{pSquare}} - {{b}} * V2f + {{a}}V2f2 ) + {{m22}} * V2f2
{{a1}} V2f2 - {{b1}} V2f + {{c1}} = 0;
V2f = {{v2fA}} OR {{v2fB}}
V1f = {{v1fA}} OR {{v1fB}}

100% Elastic Second Approach - Solve for V1f first

{{m1}} * {{v1}} + {{m2}} * {{v2}} = {{m1}} * V1f + {{m2}} * V2f
V2f = ({{m1}} * {{v1}} + {{m2}} * {{v2}} - {{m1}} * V1f) / {{m2}}
V2f = ({{p}} - {{m1}} * V1f) / {{m2}} = {{pOverm2}} - {{m1Overm2}} * V1f
12 * m1 * V12 + 12 * m2 * V2 2 = 12 * m1 * V1f2 + 12 * m2 * V2f 2
12 * {{m1}} * {{v1}}2 + 12 * {{m2}} * {{v2}} 2 = 12 * {{m1}} * V1f2 + 12 * {{m2}} * V2f 2
{{ke1}} = 12 * {{m1}} * V1f 2 + 12 * {{m2}} * V2f 2
{{ke1}} = {{m12}} * V1f2 + {{m22}} * V2f2
{{ke1}} = {{m12}} * V1f2 + {{m22}} * ({{pOverm2}} - {{m1Overm2}} * V1f))2
{{ke1}} = {{m12}} * V1f2 + {{m22}} * ({{pSquare2}} - {{bb}} * V1f + {{aa}}V1f2 )
{{a2}} V1f2 - {{b2}} V1f + {{c2}} = 0;
V1f = {{v1fA2}} OR {{v1fB2}}
V2f = {{v2fA2}} OR {{v2fB2}}

Inelastic calculation

m1 * V1 + m2 * V2 = (m1 + m2) * V2f
{{m1}} * {{v1}} + {{m2}} * {{v2}} = ({{(m1 + m2)}}) * V2f
V2f = ({{m1}} * {{v1}} + {{m2}} * {{v2}}) / ({{(m1 + m2)}})
V2f = ({{m1 * v1 + m2 * v2}}) / ({{(m1 + m2)}}) = {{inelasticV}}
KE initial = 12 * m1 * V12 + 12 * m2 * V2 2
KE initial = 12 * {{m1}} * {{v1}}2 + 12 * {{m2}} * {{v2}} 2 = {{ke1}}
KE Final = 12 * (m1 + m2) * {{inelasticV}}2 = {{inelasticKE}}
Energy Loss = {{ke1}} - {{inelasticKE}} = {{(ke1 - inelasticKE)}}