##
Special Relativity in 14 Easy (Hyper)steps

**13. Can a massive object exceed
the speed of light?**
What happens if a relativistic space craft launches a
fast probe? Can the probe travel faster than the speed of light?

Consider the following setup:

*Sulaco* launches a fast probe **with
speed u** as measured by *Sulaco*. The probe passes
*Nostromo*
which, according to *Sulaco*, is streaking to the left with speed
v.

The probe happens to nick a pair of sensors on *Nostromo*
as *Nostromo* streaks past; the flashes from these minor collisions
are noted by observers in *Sulaco's* rest frame. Not at all surprising,
*Sulaco's*
observers see that the ratio of the distance and the time interval between
the flashes is just Dx/Dt
= u. (*Nostromo* is moving, but *Sulaco's* observers know
that their probe [which travels with speed u] was at the site of the first
spark and, later, at the site of the second spark.)

From the *Nostromo's* frame, things look somewhat
different: the moving *Sulaco* (moving with speed v) launches a probe
which, in turn, moves with speed v' as shown in the figure.
If *Sulaco* is moving close to the speed of
light, and if the probe's speed
in *Sulaco's* rest frame is also close to the speed of light,
will *Nostromo* see the probe moving faster than c?

Let's use the Lorentz transformations to relate Dx
and Dt as seen by *Sulaco* (recall that
Dx/Dt = u) with *Nostromo's*
measurement of v' = Dx'/Dt'
where Dx' is the spacing between the sensors
in *Nostromo's *rest frame and Dt'
is the time interval between the flashes as measured by *Nostromo*.

You have to be careful about the sign on the velocity
to be used in the Lorentz transformations; it's positive here.

According to *Nostromo*, the probe's speed is

.

Even if the probe is very fast (so that u is almost as
big as c), and *Sulaco* is moving relativistically (so that v is almost
as big as c), this **relativistic velocity addition
formula** indicates that the probe velocity, according to *Nostromo*,
is never as large as c.

In the limit that u and v approach c,

.
Things don't go faster than c!