A very interesting concept of a head-on Elastic Collision of two
objects.
Assume that ma and mb are the masses of two objects A and B.
Let their initial velocities be ua and ub and their final velocities
va and vb.
By the conservation of momentum:
ma ua + mb ub = ma va + mb vb
or
ma (ua - va) = mb (vb - ub) ------- (1)
For elastic collision, total K.E. will conserve and
1/2 ma ua2 + 1/2 mb ub2 = 1/2 ma va2 + 1/2 mb vb2
then
ma ua2 +
mb ub2 =
ma va2 + mb vb2
ma (ua2 - va2 ) = mb (vb2- ub2 )
ma (ua - va ) (ua + va) = mb (vb-ub) (vb + ub ) ----- (2)
Put the (1) into (2), then
mb (vb - ub) (ua + va) = mb (vb-ub) (vb + ub )
(ua + va) = (vb + ub )
or
(ua - ub) = (vb - va) = - (va - vb)
(ua - ub) = - (va - vb) -----(3)
What is the meaning of the above equation (3)? Can you use it
in the supplementary exercise of Momentum and Impulse?