Suppose two bodies $A$ and $B$ with equal mass are kept at a distance. $A$ starts moving towards $B$ at a constant velocity, they hit, and I assume that it is a PERFECTLY INELASTIC collision here.
According to what I learnt from my book, suppose a body hits another body and inelastic collision takes place, in order to maintain conservation of momentum, the final velocity of the two bodies will be equal and they will stick together.
But I was thinking about it with respect to velocities and I came across a dilemma. Suppose the velocity of $A$ is $V_A$ and of $B$ is $V_B$. Before collision, $V_B=0$. After $A$ hits $B$, it applies a force on it, and $V_B$ gains some velocity. Since I assumed that both of them have equal mass, the acceleration of $B$ will be equal to retardation of $A$. Let the change in velocity at the instance of the collision be $x$.
relative velocity of $A$ w.r.t. B before collision$=V_A-V_B=V_A$
relative velocity of $A$ w.r.t. B at the instant of collision after instantaneous acceleration$=V_A-x-(V_B+x)=V_A-x-V_B-x = V_A-V_B-2x$
This shows that the relative velocity of $A$ w.r.t. $B$ has decreased.
But doesn't this mean that $A$ will apply less force at $B$ now?
$A$ will have less retardation, and $B$ will have less acceleration than before.
But doesn't that mean that the rate of decrease of velocity of $A$ as well as the rate of increase of velocity of $B$ will keep decreasing? Which means relative velocity of one of the bodies with respect to the other will keep decreasing till it approaches $0$, but will never be $0$?
So does this mean the two bodies never truly attain same velocity?
P.S: I referred to this post which I had asked long back before knowing about collisions based on a thought experiment