One of the pitfalls in current methods of acceleration analysis of planar mechanisms is the difficulty in identifying the different types of acceleration components such as the "sliding" acceleration and the Coriolis contribution. Furthermore, the latter is often missed by the analyst altogether which then leads to completely false results in dynamic analysis. Even distinguished authors have on the record actual numerical examples where the wrong angular velocity was used in computing the Coriolis component. The present article demonstrates the acceleration analysis of planar mechanisms using complex-number algebra. This technique, when programmed for digital computation using complex-arithmetic, or using hand calculation, provides the magnitude and direction of all the acceleration components, including the Coriolis term, automatically without resort to such crutches as a "rule of thumb" for determining whether or not the latter is present, "traditional sign conventions", and without the risk of using the wrong angular velocity. The procedure, derived here, is illustrated by examples.