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Capturing space targets by space robots is significant for on-orbit service and is a challenging research topic nowadays. This paper focuses on the dynamics and control of capturing a non-cooperative space target by a space robot with a long flexible manipulator. Firstly, the dynamic equation of the flexible space robot is given. The Hertz contact model is used to describe the contact force between the robot and the target. Secondly, an active compliance controller is designed to reduce the capture impact on the robot. Finally, the capture impact on the whole system is analyzed in detail in four scenarios: the combination of two kinds of movement forms and two kinds of relative positions of the robot and the target before capturing. Simulation results indicate that the capturing operation may cause complicated dynamic behaviors such as the vibration of elastic links and the continuous collision of the target. Moreover, the results show that the control method effectively offsets the capture impact on the space robot system. In general, this work lays a theoretical foundation for further study of the dynamic phenomena of the capture process.