Independently of the solution of the RP equation as written above, the difference of units between right-side terms (m^3/s² vs m/s²) results from the precedence of multiplication on “/” division (it is not trivial because it depends on both scientific communities and the use of / vs ÷):
(1/ρ*R) should be (1/(ρ*R))
In this way, every right-side terms are an acceleration (Unitful no longer complains and… a manually check is right).