Question

a 5.00 kg ball, moving to the right at a velocity of 2.00 m/s on the frictionless table, collide head on with a stationary 7.50 kg ball. find the final velocities of the balls if the collision is elastic, completely inelastic the balls stick together

Answer 1

For an elastic collision, both kinetic energy and momentum must be conserved. A good shortcut for elastic collisions is to look at it from the centre of mass reference frame and realize that the objects simply reverse velocities.

This satisfies conservation of momentum, because velocity of centre of mass is conserved.
It satisfies conservation of kinetic energy, because, in the centre of mass reference frame, each object still has its initial KE due to no change in centre of mass reference frame speed.

In the lab reference frame, ball 1 initially moves at velocity v1i, and ball 2 is initially stationary. The velocity of the centre of mass is given by the equation:
vcm = m1*v1i / (m1 + m2)

Translate from lab reference frame to centre of mass reference frame:
Use u to mean velocity in the centre of mass reference frame.
u1i = v1i - vcm
u2i = v2i - vcm, and because it was stationary, u2i = -vcm

Now, to find final velocities, swap directions:
u1f = -u1i
u2f = -u2i

Thus:
u1f = vcm - v1i
u2f = vcm

And translate back to the laboratory reference frame:
v1f = vcm + u1f
v2f = vcm + u2f

And substitute and simplify:
v1f = v1i*(m1 - m2)/(m1+m2)
v2f = 2*v1i*m1/(m1 + m2)

In the case of the inelastic collision, the balls stick together and end moving at a common velocity.
Since velocity of centre of mass is conserved in order to satisfy conservation of momentum, we've already solved our problem.

v2f_inelastic = v1f_inelastic = vcm = m1*v1i/(m1 + m2)

Given : -
m1:=5.00 kg; m2:= 7.5 kg; v1i =+2.00 m/s;

A)

v1f = v1i*(m1 - m2)/(m1+m2)

v1f = 2(5-7.5)/(5+7.5)

v1f = - 0.4 m/s.

Sign indicates it is now moving to the left in the lab reference frame

v2f = 2*v1i*m1/(m1 + m2)

v2f = 2*2*5/(5+7.5)

v2f = 1.6 m/s

Positive sign indicates it is now moving to the right in the lab reference frame

B)

v1f_inelastic = m1*v1i/(m1 + m2)

v1f_inelastic = 0.8 m/s

positive sign indicates it is now moving right in the lab reference frame

v2f_inelastic = m1*v1i/(m1 + m2)

v2f_inelastic = 0.8 m/s

positive sign indicates it is now moving right in the lab reference frame