(Solved): An experimental configuration, set up in a train, consists of a light source \( (S) \), a light de ...

An experimental configuration, set up in a train, consists of a light source \( (S) \), a light detector \( (D) \) and a mirror \( (M) \). The line \( S D \) of length \( 2 l \), joining the source and the detector, is parallel to the surface of the mirror. The line \( M N \) is normal to the mirror as shown. Two observers, \( O \) and \( O^{\prime} \), are measuring the angles of incidence \( (\alpha) \) and of reflection \( (\beta) \) in the experiment. Observer \( O \) is inside the train at rest with respect to the experimental configuration, while observer \( O^{\prime} \) is standing outside the train at rest on the railway platform. When the train is at rest, both \( O \) and \( O^{\prime} \) measure the angles \( \alpha \) and \( \beta \) to be equal. (a) (1 point) What is the relationship between angles \( \alpha \) and \( \beta \), as measured by \( O \) when the train is moving to the right with a constant speed \( v \) ? (b) (6 points) Show that the relationship between angles \( \alpha^{\prime} \) and \( \beta^{\prime} \), as measured by \( O^{\prime} \) when the train is moving to the right with a constant speed \( v \), is \[ \cos \beta^{\prime}=\frac{\cos \alpha^{\prime}\left(1+\frac{v^{2}}{c^{2}}\right)-2\left(\frac{v}{c}\right)}{\left(1+\frac{v^{2}}{c^{2}}\right)-2\left(\frac{v}{c}\right) \cos \alpha^{\prime}} . \] (c) (3 points) With the train moving to the right with a constant speed \( v \), for what value of \( v / c \) would \( O^{\prime} \) measure the angle of incidence \( \alpha^{\prime} \) and the angle of reflection \( \beta^{\prime} \) to be equal to \( \pi / 4 \) and \( \pi / 2 \), respectively?