A ball with a mass of # 2 kg# is rolling at #2 m/s# and elastically collides with a resting ball with a mass of #9 kg#. What are the post-collision velocities of the balls?

The initially moving ball would move #1.273 m/s# to the left while the ball initially at rest would move #0.72m/s# to the right.

Explanation:

(NOTE: velocities with apostrophes denotes the arbitrary object's final velocity )

Consider the total momentum of the system.

#mv_1+mv_2 = mv'_1+mv'_2 #
Note that final velocities are positive because I assumed them to move to the right after the impact.

#mv_1+mcancel(v_2) = mv'_1+mv'_2 #
#(2)(2) = (2)(v'_1)+(9)(v'_2 )#

Moreover, the problem states that the impact involves an elastic collision. Therefore, the coefficient of restitution of the system, #e#, is 1.
#e = 1=(v'_2-v'_1)/(v_1-v_2) = (v'_2-v'_1)/(2-0) #

#therefore,2 =v'_2-v'_1#

To solve the post-velocities solve the system of equations:
#(1) : 9v'_2 +2v'_1=4#
#(2) : v'_2-v'_1=2#

Where #v'_1= -1.273#
and #v'_2 = +0.72#

Since #v'_1# is negative, the assumed direction of its final velocity is the opposite.