Final answer:
The question is about calculating the time ratio and times for strokes in a quick return mechanism in Physics, likely at the college level. Without the type of mechanism specified and its design parameters, a precise answer can't be provided.
Step-by-step explanation:
The subject of this question is Physics, specifically related to the topic of rotational motion and angular kinematics. While the precise calculation of the time ratio q, the time t1 for the outward stroke, and the time t2 for the return stroke in a quick return mechanism would typically require specific details about the mechanism's design (like the length of the crank and the sliding arm), the information provided about the angular speed (Ω) and the imbalance angle (β) suggests a focus on the theoretical aspects of such systems, often encountered in engineering courses.
To solve for time ratio q, as well as times t1 and t2 theoretically, one would need to use a standard approach for the type of quick return mechanism in question (such as a Whitworth mechanism or a crank and slotted lever mechanism) and apply trigonometric relationships to determine the durations of the strokes based on the given angular speed and imbalance angle. Unfortunately, without more context or the type of quick return mechanism, providing specific times t1 and t2 is not possible.