How is the elastic collision equation derived?

Collisions are called elastic collisions if, in addition to momentum conservation, kinetic energy remain conserved too.

m1 – Mass of object 1;

m2 – Mass of object 2;
v1i – velocity of object 1 before collision;
v2i – velocity of object 2 before collision;
v1f – velocity of object 1 after collision;
v2f – velocity of object 2 after collision;

Momentum Conservation:
m1v1i+m2v2i=m1v1f+m2v2f

Rearrange this by bring all therms with m1 on one side and terms with m2 on the other side,
m1(v1i−v1f)=m2(v2f−v2i) …………… ( 1 )

m1(v1i−v1f)/m2(v2f−v2i)=1 ……….. ( 2 )

Kinetic Energy Conservation:
1/2m1v²1i+1/2m2v²2i = 1/2m1v²1f+1/2m2v2²f

Rearrange this by bring all therms with m1 on one side and terms with m2 on the other side and cancel the common factor of ‘1/2’,

m1(v21i−v21f)=m2(v22f−v22i)
m1(v1i−v1f)(v1i+v1f)=m2(v2f−v2i)(v2f+v2i)
m1(v1i−v1f)/

Recognize that the first term on the LHS is just ‘1’ [ Equation ( 2 ) ]v1i+v1f=v2i+v2f ………………… ( 3a )

v2f=v1i+v1f−v2i ……………….. ( 3b )

Substitute Equation ( 3b ) in Equation ( 1 ) to eliminate v2f

m1(v1i−v1f)=m2((v1i+v1f−v2i)−v2i)

m2(v2f−v2i).(v1i+v1f)=(v2f+v2i)

v1f=(m1−m2/m1+m2)v1i+(2m2/m1+m2)v2i

v2f=(2m1/m1+m2)v1i+(m2−m1/m1+m2)v2i ……… ( 5 )