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).

According to relativity, to a stationary observer in the garage, the ladder contracts due to its high velocity. So, briefly, it fits completely! However, from the perspective of someone riding the ladder, it’s the garage that has contracted even more!

So who’s right? The answer lies in understanding that simultaneity is relative. Events that appear simultaneous to one observer (“both doors closed at the same time”) are not for another. Both are correct in their own frame.

To see this more precisely, define two events:

• A: The front door closes at position \(x = 0\), at time \(t_A\).
• B: The back door closes at position \(x = L_{\rm garage}\), at the same time \(t_B = t_A\) (simultaneous for the garagist).

In the ladder’s frame these two doors do not close simultaneously:

$$ \begin{aligned} \Delta t' &= t'_B - t'_A = \gamma\bigl(\Delta t - \frac{v\,\Delta x}{c^2}\bigr) \\[6pt] &= \gamma\Bigl(0 - \frac{v\,L_{\rm garage}}{c^2}\Bigr) \\[6pt] &= -\,\gamma\,\frac{v\,L_{\rm garage}}{c^2} \\[4pt] &< 0 \end{aligned} $$

That is, in the ladder’s frame event B (back door) happens before A (front door).

• First the ladder’s “nose” hits the back door (B).
• Only later does the tail reach the front door (A).

There is never a moment when both doors enclose the entire ladder at once in its own frame. Hence, it never fits entirely in the garage from that viewpoint.

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Conclusion

In relativity, many seemingly absurd things make perfect sense once you understand the math. And remember: if you carry a ladder at nearly light speed, check with your insurance company first 🥸.