Start with the
relative velocity equation for a moving frame of reference:
v_{B} = v_{A} + Ω × r_{B/A} + (v_{B/A})_{rel}
Differentiate with respect to time:
a_{B} = a_{A} + dΩ/dt × r_{B/A} + Ω × dr_{B/A}/dt + dv_{B/A}/dt
Similar to velocities, the xy coordinate system is rotating inside the XY coordinate system. This means the time derivatives of the unit vectors, di/dt and dj/dt are not zero but
di/dt
= Ω × i
dj/dt = Ω × j
Thus, the time derivative of the position vector, r_{B/A} is
Putting all the terms together gives,

a_{B} =a_{A} + dΩ/dt × r_{B/A} + Ω × (Ω × r_{B/A})
+ 2 Ω × (v_{B/A})_{rel} + (a_{B/A})_{rel}


