Explain why 1-bromo-2-butene and (chloromethyl)benzene react rapidly via both SN1 and SN2 reaction pathways.

This is based on the substitution experiment and the theories behind SN1 and SN2 reactions. First challenge is to draw the structures correctly. Both systems are alkyl halides with reasonable leaving groups (halide ions).   
look at image 1
In SN2, where the Nu attacks and displaces the leaving group and the same time.  The key factor is steric hindrance. Since both these systems are primary, then the approach of the Nu is unhindered and reaction is fast.   

Image 2
In SN1, the rate determining step is the loss of the leaving group creating the  carbocation. The key factor is the stability of the carbocation.  Although both are primary (which is normally unfavourable), here both are resonance stabilised.

image 3

Here is Image 1

Here is Image 2

Here is Image 3