Fourier transform plots always show the frequency and a "mirror image" negative of the same frequency. See the Mathworks help https://www.mathworks.com/help/matlab/math/fourier-transforms.html
So your first plot indicates a single frequency.
I suggest creating some data samples from known...
I haven't checked everything, but it looks like you are saying ALL of the wheels energy converts to ball kinetic energy? Is the motor on or off during launch? What is the wheel speed after the ball is launched?
I am trying to think of the usefulness of the information in the gray graph - average speed of remaining trip.
This is not like predicting how much time is remaining. You calculate only after you know the total time, if I understand you correctly.
See this video for an interesting demonstration and working out for the acceleration in a circular path.
Note, for a circular path, the acceleration direction is changing - it points toward the center of the circle as the object moves.
Friction actz on the skate blade. So the boy can change...
Acceleration and Force are vectors with both magnitude and direction. In a circular path, the magnitude of the acceleration is constant, but the direction is always toward the center of the circle. So this is not constant acceleration.
This is from 2007. I'm not sure why it is popping up now... The OP has answered its own questions. It looks correct except for a misplaced parenthesis in part c: at^(2/2) should be a(t^2)/2 and correct units would be helpful.