In this article, we propose a new method for multi-robot systems to have reactive responses to various collision scenarios in real time. This method contains a novel p-norm approximation-based reactive approach, which allows multiple robots to avoid mutual collisions or to adapt to the collisions with intelligently regulated force. Compared with existing approaches, the implementation of the proposed method is simpler and more convenient since our reactive approach works as several linear Quadratic Programming (QP) constraints, allowing for flexible utilization by Inverse Kinematics (IK) solvers. In addition, it requires low computational complexity, achieves high accuracy, and does not need training. In the experiments, we employ a multi-robot system to conduct comprehensive comparisons between the proposed method and state-of-the-art approaches, effectively showcasing the efficacy of our work.