Final answer:
To find dy/dt, or velocity, we use integral calculus to work backward from acceleration functions, considering directionality for motion analysis.
Step-by-step explanation:
To find dy/dt for each pair of functions, we first need to understand the relationship between position, velocity, and acceleration. The velocity function v is the time derivative of the position function s, written as v = ds/dt. Similarly, acceleration a is the time derivative of the velocity function, written as a = dv/dt. When given an acceleration function that depends on time, we can directly integrate this function to find the velocity function, and then integrate the velocity function to find the position function.
For the case when acceleration is a function of displacement, we have the relation ads = udv, which relates acceleration and displacement. By using integral calculus, starting with acceleration and integrating backwards, we can find the velocity and position functions. When comparing directions of velocity and acceleration, it is important to note that they can be in the same or opposite directions, affecting the motion of the object.