Background: Recognizing the relatively few studies that have explored university students' understanding and alternative conceptions in linear and circular motion as well as the relatively few studies that have used 2-dimensional multiple-choice questions (MCQs) in the science education literature, this study set out to contribute to these areas. Purpose: To investigate undergraduates' understanding of velocities and accelerations in linear and circular motions using 2-dimensional MCQs. Sample: 450 engineering undergraduates who were doing a Physics course. Design and methods: A 2-dimensional MCQ instrument comprising 8 questions was developed and administered to the samples. Both cognitive scores and metacognitive measures were extracted as part of data analyses. Incorrect responses which were at least at the 10% threshold level were classified as alternative conceptions (ACs). Results: Both cognitive scores and metacognitive measures have afforded a more nuanced appreciation of students' understanding as well as their ACs on the topics. In this context, we report on confidence when correct, confidence when wrong, confidence with which the ACs are expressed and confidence accuracy quotient for each question. Student responses led to the identification of 14 ACs. It was noted that students' understanding of key concepts in the topics tested is rather modest. A commentary is presented on these ACs from the lens of instructor teaching experience. Conclusion: Two dimensional MCQs were found to be effective in exploring undergraduates' understanding of linear and circular motions as well as in documenting their alternative conceptions.