Two spheres collide and assume that the collision is perfectly elastic

[nullspace7]

Two spheres collide and assume that the collision is perfectly elastic. Also --only linear momentum.

I have the relationship:

(va' - vb') dot N = -(va - vb) dot N

Where N is the normal vector at the point of collision. va and vb are initial velocities of object A and B, respectively. And va' and vb' are the final velocities of object A and B respectively.

I want to know how this relationship is derived.

This is what I try:

Relative to object B, object A has velocity v_ab. Relative to object B,
object B has velocity 0.

m_a*v_a + m_b*v_b = m_a*v_a' + m_b*v_b'

Relative to B:

m_a*v_ab + 0= m_a*v_ab' + 0

v_ab = v_ab'

That would give me this: (va' - vb') dot N = (va - vb) dot N

But I am missing the negative sign, because they should be opposite. Please
advise. Thanks in advance.

 

[geometer]

As I'm sure you're aware, in an elastic collision, both kinetic energy and momentum are conserved. If you write both sets of equations and solve them simultaneously, you'll get (va' - vb') = - (va - vb)

 

[nullspace7]

Yeah (needed KE), I've solved it now. I solved it in the center of mass frame, which seemed easier.