A4 kg cart with some unknown velocity has a force acting on it resulting in 110J of work being added to the system. If the cart's final velocity is 8.0 m/s,
determine the initial velocity of the cart.

Given momentum is conserved and is mv

if
mass is the same and va=2vb
then momentum is m(2vb + vb)

if collision is elastic (bounces off)
then m(2vb + vb) = m(vfa + vfb)
3vb = vfa + vfb
meaning final velocities would be different

if collision is inelastic (sticks together)
then m(2vb + vb) = (m+m)vf
meaning final velocity would be same

is this what you wanted?