Velocity Addition Calculator
Calculate relativistic velocity addition using Einstein's special relativity formula
Calculate Relativistic Velocity Addition
Velocity of spaceship as seen by stationary observer
Velocity of projectile relative to spaceship
Velocity Addition Results
Relativistic Velocity (u)
Classical Velocity (v + w)
Relativistic Formula: u = (v + w) / (1 + vw/c²)
Classical Formula: u = v + w
Relativistic Factor: 1.000000 (ratio of relativistic to classical)
Spaceship and Laser Example
Scenario
Spaceship velocity: v = 0.6c (60% speed of light)
Laser beam: w = 1.0c (speed of light relative to spaceship)
Question: What speed does a stationary observer measure for the laser?
Calculation
u = (v + w) / (1 + vw/c²)
u = (0.6c + 1.0c) / (1 + (0.6c × 1.0c)/c²)
u = 1.6c / (1 + 0.6) = 1.6c / 1.6
Result: u = 1.0c (exactly the speed of light!)
Classical prediction: u = 0.6c + 1.0c = 1.6c (impossible!)
Key Insights
Speed Limit
No matter how you add velocities, the result never exceeds the speed of light.
Low Speed Approximation
At everyday speeds (v, w ≪ c), the relativistic formula reduces to simple addition.
Light Speed Invariance
If either velocity equals c, the result is always c, regardless of the other velocity.
Example Velocities
Everyday Speeds
Car: ~0.00000003c
Bullet: ~0.000001c
Space probe: ~0.00006c
High Energy Physics
Electron in accelerator: ~0.999c
Cosmic ray proton: ~0.99999c
Neutrino: ~0.999999c
Theoretical Examples
0.5c + 0.5c = 0.8c
0.9c + 0.9c = 0.994c
0.99c + 0.99c = 0.99995c
Understanding Relativistic Velocity Addition
Why Can't We Just Add Velocities?
In everyday life, velocities add simply: if you walk forward on a moving train, your speed relative to the ground is the sum of the train's speed and your walking speed. However, this breaks down at high speeds due to the fundamental nature of space and time.
The Speed of Light is Special
Einstein discovered that the speed of light is the same for all observers, regardless of their motion. This seemingly simple fact has profound consequences for how velocities combine at high speeds.
Real-World Applications
- •Particle accelerators must account for relativistic effects
- •GPS satellites use relativistic corrections
- •Cosmic ray interactions in the atmosphere
The Mathematics
u = (v + w) / (1 + vw/c²)
Einstein Velocity Addition Formula
- u: Resultant velocity (what stationary observer sees)
- v: Velocity of moving reference frame (spaceship)
- w: Velocity relative to moving frame (projectile)
- c: Speed of light in vacuum
Key Point: When v or w equals c, the denominator ensures that u = c, preserving the universality of the speed of light.
Low Speed Limit
When v, w ≪ c, the term vw/c² ≈ 0, so u ≈ v + w (classical addition).
High Speed Effects
As velocities approach c, the denominator grows, preventing u from exceeding c.
Light Speed Invariance
If v = c or w = c, then u = c regardless of the other velocity.