According to special relativity, the relative velocity between two objects moving in opposite directions cannot simply be calculated by adding their velocities. Instead, we need to use the relativistic velocity addition formula.
The relativistic velocity addition formula is given by:
v' = (v1 + v2) / (1 + (v1 * v2) / c^2),
where v' is the relative velocity between the two objects, v1 is the velocity of one object, v2 is the velocity of the other object, and c is the speed of light in a vacuum (approximately 3.00 × 10^8 meters per second).
Let's calculate the relative velocity observed by A and the stationary observer separately:
- Relative velocity observed by A: Since the velocity of particle A is 0.9c and the velocity of particle B is also 0.9c in the opposite direction, we can substitute these values into the relativistic velocity addition formula:
v' = (0.9c + (-0.9c)) / (1 + (0.9c * -0.9c) / c^2) = 0 / (1 + (-0.81)) = 0 / (0.19) = 0.
Therefore, as observed by particle A, the relative velocity between A and B is 0.
- Relative velocity observed by the stationary observer: In this case, the velocity of the stationary observer is 0 since they are at rest. Substituting v1 = 0 and v2 = 0.9c into the relativistic velocity addition formula:
v' = (0 + 0.9c) / (1 + (0 * 0.9c) / c^2) = 0.9c / (1 + 0) = 0.9c.
Thus, as observed by the stationary observer, the relative velocity between A and B is 0.9c.
To summarize:
- As observed by particle A, the relative velocity between A and B is 0.
- As observed by the stationary observer, the relative velocity between A and B is 0.9c.