##
Special Relativity in 14 Easy (Hyper)steps

##
**4. Q**uantitative description
of time dilation

Let's calculate
how much longer it takes a moving light clock
to tick, compared to a stationary
light clock.
The idea is simple:
light **always** travels
about a foot
per nanosecond.
The light beam inside a moving light clock travels farther per "tick" than
the the light beam
inside the stationary clock. We can determine the time between ticks once
we know the distances involved.

The bottom-mirror-to-top-mirror
trip
only takes t = D/c in the stationary clock.

Let's say the corresponding trip takes t'
in the moving clock. During time t', the clock moves
to the right a distance vt'. The light beam travels along the
hypotenuse
of right triangle whose other sides have lengths vt' and D.

Since light always travels at c, we
must have .

We can solve for t' by squaring both sides and doing
some algebra. The result is .

Take a look
at what we've found: the **moving** clock
takes a factor of
longer to tick than does the **stationary**
clock, since its light beam travels farther per tick.

When
the moving clock can take a **very long **
time
to tick compared to the stationary clock. Moving clocks (and
**everything
else** that's moving with respect to an observer) seem to be running
too slowly!

Fast-moving
people think slowly, respond slowly, get jokes
slowly,... relative to the observers who watch them.