Building a technology-driven world appears to be the main motivational force behind students choosing to undertake engineering studies. The first year of engineering education plays a significant role in demonstrating sufficient mathematical and scientific rigor to satisfy these motivational factors. The common applied mechanics courses play a central role in achieving this. At the same time, a vast majority of students suffer from a lack of the necessary mathematical skills and analytical orientation for various reasons. Due to different educational philosophies and teaching pedagogies, a lack of proper integration between mathematics and applied mechanics is common. Several efforts were made to build better curriculum, teaching, and learning systems, resulting in widely varied solutions, but most of them require drastically different implementation approaches. With sufficient rigor in teaching and assessment, the first-year applied mechanics (common) courses designed for engineering students can solve students' mathematical and motivational lapses and help bridge the gaps between pre-university and university education endeavours. This paper presents evidence supporting this argument. In particular, datasets collected from the direct experiences delivering the first-year static and dynamics courses to many students over the past decade and a half are analysed to establish the proposition.