13.
Two reference frames S and S’ are moving towards each other with relative velocity 3c/4,
where c is the speed of light. Now an object in S frame is moving with velocity 3c/4
towards the same direction that S frame is moving. What will be the velocity of that
object as measured from S’ frame ?
a. c
b. 3c/4
c. 3c/2
d. 24c/25
Relativistic velocity addition governs how speeds combine in special relativity when reference frames move relative to each other. Frames S and S’ approach with relative speed 3c/4, meaning S’ moves at -3c/4 (leftward) relative to S along the x-axis. The object moves at +3c/4 (rightward) in S, in the direction S moves relative to S’.
Velocity Addition Formula
The relativistic formula for parallel velocities is u′ = (u – v) / (1 – uv/c²), where v = -3c/4 is S’ velocity relative to S, and u = 3c/4 is the object’s velocity in S. Substituting values (c=1): numerator 3/4 – (-3/4) = 6/4, denominator 1 – (3/4)(-3/4) = 1 + 9/16 = 25/16. Thus, u′ = (6/4) / (25/16) = (6/4) × (16/25) = 24/25 c.
Option Analysis
Classical addition would give 3c/4 + 3c/4 = 3c/2, exceeding c, which violates relativity.
| Option | Value | Explanation |
|---|---|---|
| a. c | c | Speed cannot reach exactly c for massive objects; result is less than c. |
| b. 3c/4 | 0.75c | Matches neither classical (1.5c) nor relativistic addition. |
| c. 3c/2 | 1.5c | Classical sum, impossible as it exceeds c. |
| d. 24c/25 | 0.96c | Correct relativistic result, approaching but below c. |
Why Relativistic Formula Matters
Velocities do not add linearly at high speeds due to time dilation and length contraction. The formula ensures no speed exceeds c, unlike Newtonian physics. Here, mutual approach at 0.75c each yields 0.96c observed speed, demonstrating velocity saturation near c.
