For the blocks \(A\) and \(B\) \(FBD\) as shown below

Equations of motion

\(a_A=\frac{F}{M}(\text { in }-x \text { direction })\)

\(a_B=\frac{F}{M}(\text { in }+x \text { direction })\)

Relative acceleration, of A w.r.t. B,

\(a_{A, B} =a_A-a_B=-\frac{F}{m}-\frac{F}{M}\)

\(=-F\left(\frac{M+m}{M m}\right) \text { (along }-x \text { direction) }\)

Initial relative velocity of Aw.r.t. \(B , u _{ AB }= v _0\) using equation \(v ^2= u ^2+2 a\)

\(0= v _0^2-\frac{2 F ( m + M ) S }{ Mm } \Rightarrow S =\frac{ Mmv _0^2}{2 F ( m + M )}\)

i.e., Distance moved by A relative to \(B\)

\(S _{ AB }=\frac{ Mmv _0^2}{2 F ( m + M )}\)