Question

In the scenario above, the 1-unit mass trolley and the 2-unit mass trolley are moving at a speed of 6 m/s in opposite direction for an head on collision. What is the minimum speed that the 1 unit-mass trolley will move after the collision?

Answer 1

You didn't tell whether its a totally inelastic collision or totally elastic one. I'll assume both one by one.
here \[m _{1}= 1 unit \]\[m _{2} = 2 unit \]\[v _{1}=6 m/s\]\[v _{2}=-6 m/s\]
In an inelastic collision\[p _{1}-p _{2}=m _{1}v _{1}-m _{2}v _{2}=6 Ns\]Let the the joint mass of both the bodies be called \[m _{3}\]\[p _{1}+p _{2}=p _{3} \]\[p _{3}= -6Ns = 3 unit mass \times -2 m/s\] Therefore in inelastic collision \[v _{3}= -2m/s\]
In Elastic collision
according to the formula \[v \prime _{1} =[(m _{1}-m _{2})\div( m _{1}+m _{2})]\times v_{2}\]\[v \prime _{1}= [(1-2)\div3]\times6= -2 m/s\]
Although the velocities were same here in both inelastic and elastic collisions, its not the same always.