In case 1 the A travels the distance D while traveling from X to Y.
In case 2 the velocity of A according to Sam will 'a' and distance travelled by A will be greater than D because the wall Y is also moving away. According to wall X velocity of ball is a-v and distance travelled by A will be D because Frame of reference of X is same as Case 1.
My question is in Case 2 how can two observer Sam and wall X measure different distances? Is it possible?
@Nitaa I would argue you can measure distances even in a moving frame; actually it is the basic concept behind Galilean realativity.
But: If you measure an object's position at two different times, then it clearly depends on its velocity relative to you. So yes, the distance travelled over a span of time is clearly relative to the observer.
In Galilean approximation, we may speak of a single universal time for all frames of reference. If measured at one single time, the wall distance will be independent of your frame of reference no matter if you move or not. There is no snag.
This result will be manifestly different in Einstein's relativity (where "universal time" does not exist).
You are essentially mixing two things together. You should meassure distance in a standing frame. In your case observer in case 2 wouldn't say that the length is time times velocity a, but he would use time times velocity a mínus velocity b. So in case 2 this is true $D =(a-v) t.
Why? Consider being in a train and meassuring length of a widnow. You would put a meter next to the window and say it measures 1 window length. You wouldn't say it measures 1 window length + the distance the train travelled during the time of measurement.
