Created by: roberto.c.alfredo in physics on Jul 8, 2025, 12:36 AM
In classical physics, adding velocities is easy. If a Mustang 🐎 travels at 100 km/h and throws a ball forward at 20 km/h (relative to the car), a stationary observer sees the ball at 120 km/h. Simple, right? But when Einstein entered the scene, these comfortable sums ended.
Why Don’t Velocities Add Normally?
Special relativity forces us to use a different formula at speeds near the speed of light:
$$ u = \frac{u' + v}{1 + \frac{u'v}{c^2}} $$
- The left-hand \(u\) is the resulting velocity, measured in the stationary frame.
- \(u'\) is the velocity of the object in the moving frame.
- \(v\) is the velocity of the moving frame relative to the stationary one.
- \(c\) is the speed of light (299,792,458 m/s).
This formula ensures that you never exceed the speed of light, preserving the universe’s consistency (phew 😅).
For example, if a spaceship travels at \(0.8c\) and launches a probe forward at \(0.7c\) relative to it, the speed measured from outside is NOT \(1.5c\). It is:
$$ u = \frac{0.7c + 0.8c}{1 + \frac{(0.7c)(0.8c)}{c^2}} = \frac{1.5c}{1 + 0.56} \approx 0.96c $$
Fast? Yes. Faster than light? Never!
The Ladder Paradox 🪜🏠
This relativistic logic gives rise to amusing and perplexing situations, like the ladder paradox (also known as the “barn paradox”).
Imagine a ladder too long to fit in your garage. Normally, you can’t squeeze it in, right? But now make the ladder race toward the garage at high speed (yes, a fast ladder—physicists can be crazy).