Here would be the thought process:
1) Secondary carbon. Can’t rule anything out at this stage.
2) Identity of nucleophile/base: Br(-). Good nucleophile (negatively charged), weak base. Rule out E2 because E2 needs a strong base. Rule out SN1/E1 because they operate with weak (generally neutral) bases. That leaves SN2.
3) So draw product of SN2 with inversion of configuration.

For an Sn1 imagine a 5 carbon chain with carbon 2 has wedge of Br and ladder of H with nucleophile of Cl- how can you tell if its a SN1 reaction. like that what I’m having problem with. please let me know asap i have my exam on Thursday!

You might want to look into the subject of “double inversion” :-) especially in basic conditions.

Primary alkyl halide. Almost certainly an SN2 reaction.

Acid-base reactions on *heteroatoms* are fast, but that rule of thumb doesn’t really apply to the E2 reaction, where you’re breaking a C-H bond with accompanying change of hybridization on the carbon…

Well, these are the kinds of things that you have to weigh when evaluating these questions. Read on to parts 2 and 3.

The halogen would not have significant pi bond character. The answer is that the resulting carbocations would be very unstable.

Primary allylic, yes. Primary alkyl halide, formation of the primary carbocation as an intermediate is unlikely. It’s possible to have concerted rearrangement reactions where migration of a neighboring group accompanies displacement of a leaving group, which is generally how these types of things happen.