You add velocities differently depending on whether you're dealing with everyday speeds (classical mechanics) or speeds approaching the speed of light (relativistic mechanics).
Classical Velocity Addition
For everyday speeds, velocities simply add together as you'd expect. This is based on the Galilean transformation. The formula is:
u = v + u'
Where:
- u is the velocity of an object relative to a stationary observer.
- v is the velocity of a moving observer relative to the stationary observer.
- u' is the velocity of the object relative to the moving observer.
Example:
Imagine you're standing on the ground (stationary observer). A train is moving at 30 m/s (v). Inside the train, someone is walking towards the front at 2 m/s (u'). To you, the person on the train is moving at 30 m/s + 2 m/s = 32 m/s (u).
Relativistic Velocity Addition
When speeds become a significant fraction of the speed of light (approximately 3 x 108 m/s), classical velocity addition no longer holds true. This is due to the principles of special relativity, which state that the speed of light in a vacuum is the same for all inertial observers. The formula for relativistic velocity addition is:
u = (v + u') / (1 + (vu'/c2))
Where:
- u is the velocity of the object relative to the stationary observer.
- v is the velocity of the moving observer relative to the stationary observer.
- u' is the velocity of the object relative to the moving observer.
- c is the speed of light.
Example:
Imagine a spaceship moving at 0.75c (v) relative to Earth. Inside the spaceship, another object is launched forward at 0.75c (u') relative to the spaceship.
Using classical velocity addition, the object's speed relative to Earth would be 1.5c, which is impossible since nothing can exceed the speed of light.
Using relativistic velocity addition:
u = (0.75c + 0.75c) / (1 + (0.75c * 0.75c / c2))
u = (1.5c) / (1 + 0.5625)
u = (1.5c) / (1.5625)
u = 0.96c
So, according to special relativity, the object's speed relative to Earth is 0.96c, which is less than the speed of light.
Key Differences Summarized
| Feature | Classical Velocity Addition | Relativistic Velocity Addition |
|---|---|---|
| Speed Range | Everyday speeds | Speeds approaching the speed of light |
| Formula | u = v + u' | u = (v + u') / (1 + (vu'/c2)) |
| Speed Limit | No limit | Speed of light (c) is the limit |
| Accuracy | Accurate at low speeds | Accurate at all speeds |
In summary, the method of adding velocities depends entirely on the speeds involved. For ordinary speeds, simple addition works. For speeds approaching the speed of light, you must use the relativistic formula to ensure that no speed exceeds the ultimate cosmic speed limit: the speed of light.
