Apparent length of the beacons line · RMPH 2013 #2
Physics
Problem
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. 2 - Apparent length of the beacons line
The camera takes a picture of the instant when the actual position of the center of beacons line is at some x0 point.
2.a. Determine the expression for the apparent length of the beacons line on this picture.
2.b. Indicate – using a mathematical expression – how the apparent length of beacons line changes with time, when SS Enterprise moves along the Ox axe. The space ship comes from far away, passes through the origin of the system and then receding far away.