Search engines operate under a strict time constraint as a fast response is paramount to user satisfaction. Thus, neural re-ranking models have a limited time-budget to re-rank documents. Given the same amount of time, a faster re-ranking model can incorporate more documents than a less efficient one, leading to a higher effectiveness. To utilize this property, we propose TK (Transformer-Kernel): a neural re-ranking model for ad-hoc search using an efficient contextualization mechanism. TK employs a very small number of Transformer layers (up to three) to contextualize query and document word embeddings. To score individual term interactions, we use a document-length enhanced kernel-pooling, which enables users to gain insight into the model. TK offers an optimal ratio between effectiveness and efficiency: under realistic time constraints (max. 200 ms per query) TK achieves the highest effectiveness in comparison to BERT and other re-ranking models. We demonstrate this on three large-scale ranking collections: MSMARCO-Passage, MSMARCO-Document, and TREC CAR. In addition, to gain insight into TK, we perform a clustered query analysis of TK's results, highlighting its strengths and weaknesses on queries with different types of information need and we show how to interpret the cause of ranking differences of two documents by comparing their internal scores.