Question

Consider an inertial frame \( S' \) moving at speed \( c/2 \) away from another inertial frame \( S \) along the common x-axis, where \( c \) is the speed of light. As observed from \( S' \), a particle is moving with speed \( c/2 \) in the \( y' \) direction, as shown in the figure. The speed of the particle as seen from \( S \) is: 

• A. \( c/\sqrt{2} \)
• B. \( c/2 \)
• C. \( \sqrt{7}c/4 \)
• D. \( \sqrt{3}c/5 \)

Hint:

For adding velocities in special relativity, use the relativistic velocity addition formula.

Answer 1

The Correct Option is A

Step 1: Understanding the relativistic velocity addition formula.
In relativity, the velocity of the particle as observed from \( S \) can be obtained using the relativistic velocity addition formula: \[ v = \frac{v' + v_{S'} }{1 + \frac{v'v_{S'}}{c^2}}. \] Here, \( v' = c/2 \) and \( v_{S'} = c/2 \) are the speeds of the particle and the frame \( S' \), respectively. Substituting these values gives the speed of the particle as seen from \( S \) as \( c/\sqrt{2} \).
Step 2: Conclusion.
Thus, the correct answer is option (A) because the speed is \( c/\sqrt{2} \).