Which is proper time $(t_0)$ and improper time $(t')$? A lighthouse has two blinking spotlights 0.4 meters apart. The spotlight on the right blinks 5 seconds after the one on the left. How much time elapses between the blinkings of the spotlights as observed by a drone traveling at 0.2c?
To solve the above question, will the 5 seconds be the improper time $(t')$ in the formula 
$$t' = \frac {t_0}{ \sqrt {1 - \frac {v^2}{c^2}}} \ \ \ ? $$
 A: This means, 5 seconds will be improper time(t') right?
But, how this can happen since the drone will see the lighthouse moving? Does this mean the time will dilated?
A: Proper time is always defined between some pair of events that both occur along some particular object's worldline (the fact that they both occur 'along' it just means the object is passing right next to each event when it happens). The physical interpretation is that it's the time that would be measured by a clock that remains next to the object, so the clock has the same worldline (the same function for its position coordinate as a function of coordinate time in any given inertial frame). And for any pair of events in special relativity, as long as the spatial separation between them in light-seconds is smaller than the temporal separation in seconds, you can always imagine a clock that is moving at the right velocity so that it's passing next to each event when it happens, and then the time dilation equation will relate the proper time between the events as measured by that clock (which is the same as the coordinate time in the clock's rest frame, where the events both happen at the same coordinate position) to the coordinate time between these same events in some other inertial frame where the clock is in motion.
In this case, I would assume that when they say "The spotlight on the right blinks 5 seconds after the one on the left", they mean that 5 seconds would be measured in the rest frame of the lighthouse, but since it occurs at two different positions in this frame (0.4 meters apart), it cannot technically be the proper time on a clock at rest in this frame. If you imagine a clock moving 0.4 meters every 5 seconds, or 0.4/5 = 0.08 meters/second, then both light-blinks could occur along its worldline. 0.08 meter/second is such a small fraction of light speed that the time it would measure between these events would only differ by a negligible amount from the time as measured in the light house frame, so it's fine to say its proper time is 5 seconds as well, and then to figure out how much coordinate time elapses between the spotlight blinks in the drone frame using the time dilation equation.
