Oblique Elastic Collision between 2 bodies
From conservation of energy, m1u12 +m2u22
=m1v12 +m2v22 ; Multiplying by m1 and
rearranging, m12(u12-v12)=m1m2(v22-u22) From conservation of momentum, m12u12 +m22u22 +2m1m2u1u2cosA =m12v12 +m22v22 ;+2m1m2v1v2cosφ or m12(u12 -v12)+2m1m2u1u2cosA = m22(v22 -u22) +2m1m2v1v2cosφ or m1m2(v22 -u22)+2m1m2u1u2cosA = m22(v22 -u22) +2m1m2v1v2cosφ or av22 -2v1v2cosφ =au22 -2u1u2cosA where a=(m1-m2)/m1 =1-m where m=m2/m1 Let v1=xv2 av22 -2xv22cosφ =au22 -2u1u2cosA=C or v22(a-2xcosφ) =C ------(1) now m1u12 +m2u22 =m1v12 +m2v22 ; v12=u12 +m(u22-v22) =(u12+mu22) -mv22 =Z -mv22 = x2 v22 ; here Z=(u12+mu22) or v22=Z /(m+ x2) Putting this value in eqn 1, (a-2xcosφ)*[Z /(m+ x2)] =C or Cx2 +2zcosφ x+(Cm-Za) =0 or Cx2 +2zcosφ x+R =0 where R=(Cm-Za) x=-(-Z/C)cosφ +√ [(Z2/C2)cos2φ -R/C] or x=-(-Z/C)cosφ -√ [(Z2/C2)cos2φ -R/C] Example : take m1=4,m2=3,u1=5,u2=8,A =30, φ =20 and try with other combinations |