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.

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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!

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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).