The link between image and the actual position · RMPH 2013 #1

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

\vec{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 \times 10^8 \text{m} \cdot \text{s}^{-1}

. Employ the quantities

\beta = v/c

and

\gamma = 1/\sqrt{1-\beta^2}

, if they help to simplify your result. Task No. 1 - The link between image and the actual position The actual position is the position in the frame in which the camera is at rest. Next, the camera remains stationary in the point

(0,d)

and the SS Enterprise is moving at velocity

\vec{v}

on Ox direction. In an image provided by the camera described above is observed that one of the bright beacons is placed in the position

x_i

. 1.a. Determine the expression for the actual position of the bright beacon when the image is formed in the camera. 1.b. Find also the corresponding inverse relation, that is the expression of

x_i

in terms of

x,d,L,v

and

c

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