Problem 1)

Let's call the inertial system of the observer on shore S and the rest system of the sailboat S'. S' is moving with velocity v=c/2 in the positive x-direction, and the mast is assumed to be in the x-y plane (as is the laser beam). Let L be the total length of the mast. Finally, let's put the origin of S' in the anchor point (the bottom end) of the mast, at some time t'=0 (and therefore at all times t').The coordinates of the top end of the mast in S' are then
x' = -L sin30o and y' = L cos30o (independent of time t', of course). On the other hand, the laser beam is propagating with velocity ux' = -c sin30o and uy' = c cos30o in S'.

How can we describe the same situation in S? The easiest way for the angle is to use the fact that "moving meter sticks are shortened by a factor 1/g". In this case, the term "moving meter sticks" means any dimension of the sailboat in the x-direction, e.g., the length of the sailboat itself, but also the projection of the mast onto the x-direction. Therefore,
Dx = x'/g = -L sin30o.(3/4)1/2 . On the other hand, all y-dimensions are unchanged, and

Dy = y' = L cos30o . For the unknown apparent angle a of recline as observed in S we get thus tan a = Dx/Dy = -tan30o.(3/4)1/2 = 1/Ã3.(3/4)1/2 = 1/2 => a =26.57o .

The mast seems somewhat less reclined than in S', but the difference is small.

On the other hand, the light beam travels with the same speed c in S as in S'. The component in x-direction can be determined from the velocity addition rule:

ux = (ux' + v)/(1 + ux'v/c2) = (-c/2 + c/2)/(1 + ux'v/c2) = 0 !

This means that the light ray must be going up vertically in S, quite different from its angle in S' (and from a).

You can convince yourself that this result makes sense by observing that the light ray takes some time dt = L cos30o/c to travel to the height of the top of the mast (in S), during which time the ship has moved by a distance dx = c/2 dt = L cos30o/2. This distance is equal to the difference Dx between the bottom and the top of the mast in S, i.e., a light signal emitted at the bottom of the mast reaches the top just as it passes the same x-position as the bottom had at emission time. (Of course, the light ray in S' has to reach the top of the mast by design - emission angle and angle of recline are the same).