If I project a flashlight beam on a wall and turn the source, the projection (as seen on the wall) is seen moving along the direction of the turn with some speed. If I have projected it on a gigantic screen say a million miles in width and turn the source quickly, can the projection then travel faster than the speed of light along the screen?
Yes, projection can move faster than light. Assuming a circular wall, and projector at the center, the maximum limit will be 2 times c.
However, there is nothing moving faster than c. Different photons are moving from projector to the wall at c. Just because you rotate the projector quickly, the projection seem to move from one side to other, but actually photons are moving from projector to the wall at different angles, at different times.
Assuming the projector can be rotated 180 degrees in no time, the projection can move from one side to the other in half the time as compared to the time taken by actual light to move from one side to the other. This is assuming initial and final projections are equidistant from the projector.
You can say, if the projector is rotated in no time, then the projection is not actually moving. No time means non-zero, very small time as compared to the time taken by light from projector to the wall.
If you make the projection move faster than 2c (by making the wall not circular, or by making final point much closer to the projector as compared to the initial point, then it will not be possible to see the projection moving continuously. In that case, you will see the final projection before the moving projection completes its journey along the wall.