Description of the situation:
Star Ship Enterprise has on each of the two warp engines a succession of five collinear, equidistant, numbered, beacons as shown in the figure (for the propeller on right side). When traveling, SS Enterprise moves with constant speed v so that, for the questions that follow, beacons located on a propeller are always on the same line that is Ox axis of the coordinate system. In the situation room of another spacecraft which is at rest and which oversees the SS Enterprise, are analyzed the images of line of beacons. The images are taken by a camera with the aperture located at the point (0,d) in the orthogonal coordinate system shown in the inset in figure above. An image of line of beacons is produced by light rays that arrive simultaneously at camera aperture that opens for a short time. On a picture of SS Enterprise at rest, the length of line of beacons is L. In the problem, the beacons can be considered as points.
The speed of light is c=3.00×108m⋅s−1.
Employ the quantities β=v/c and γ=1/1−β2, if they help to simplify your result.
Task No. 3 - Symmetrical image
One of the pictures taken by the camera shows both ends of beacons line at the same distance from the aperture of camera.
3.a. Determine the expression for the apparent length of beacons line on this picture.
3.b. Determine the expression for the actual position of the middle beacons at the time when this picture is taken.
3.c. Determine where will be situated on the picture the image of the middle beacons.