Final answer:
To determine the time path for difference equations, it is essential to consider both the mathematical solutions and their physical relevance, selecting only the positive time values that make sense within the context of the equation's application.
Step-by-step explanation:
The student is looking to find the time path for a set of different equations. Various examples given deal with finding the correct value of time in situations involving motion and kinematics, indicating the importance of considering only physically meaningful solutions, such as positive times, since negative times suggest events that happened before the motion began and are thus not reasonable.
For instance, when working with equations of motion, an equation that provides two solutions for time, such as t = 10.0 s and t = -20.0 s, necessitates disregarding the negative solution as it is not feasible. Similarly, with a quadratic equation like 1² + 1 - 20 = 0 yielding roots t = -5.0 s and t = 4.0 s, the positive root t = 4.0 s is the one to consider since it represents a point in time after the motion has started. These examples showcase the process of examining each potential solution and selecting the one that aligns with the physical context of the scenario.