Natural Science Forum / Physics / Relativity / January 2005

                   Sam, Joe, a mosquito and a ladder.
                                  by Androcles

Much of this story is credited to Daryl McCullough, only the ladder
was added by me.  It explains the origins of Einstein's Special
Relativity
for those having difficulty grasping the subject.

Sam and Joe are housepainters, and are walking along the street at 3 fps
in still air carrying a 32 ft long ladder between them, Joe leading the
way. Sam is carrying some paint cans and Joe has the brushes and
rollers.

At some point along their journey a mosquito name Albert buzzes past
Sam's
ear. Sam swats at it, but drops a can of red paint as he does so.

Albert flies along the ladder from Sam to Joe at a constant speed
of 5 fps. When it reaches Joe, Joe also swats at it, but drops a paint

roller. Albert, still hungry but not liking the smell of Joe's cigar,
flies back along the ladder toward Sam, again with a constant speed of
5 fps in the still air. Upon reaching Sam, once again Sam tries to swat
the
wee beastie but drops a can of green paint. He yells as the mosquito
bites
him and this startles Joe, who drops a paint brush.

Now it's your turn. I'll give the answers further down, but take a
moment
to do the calculations for yourself.

1) How many seconds did it take for Albert to fly from Sam to Joe?
2) How many seconds did it take for Albert to fly from Joe to Sam?
3) How far is it between the red paint can and the roller?
4) How far is it between the green paint can and the roller?

(Answers below)
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Assume the speed of the mosquito is c = 5 fps.
The speed of Sam and Joe is v = 3 fps, given.

We then must have a distance along the road for Joe of
32ft + vt, and  for the mosquito, a distance of ct.

Solving for t,
ct = 32 + vt
ct - vt = 32
t(c-v) = 32
t = 32 /(c-v) = 32/(5 - 3) = 16 seconds
So the answer to  Q.1) is 16 seconds.

The mosquito coming back is going to meet Sam going forward,
so it flies along the 32 feet of the ladder in time
t = 32/(c+v) = 32/8 = 4 seconds.

The answer to Q.2) is therefore 4 seconds.

The distance from the dropped red paint can to the dropped roller
is just ct, or 5 * 16 = 80 feet, so the answer to Q.3) is 80 ft.
Or we could do it by vt + 32 = 3 * 16 + 32 = 80, once again.
(Remember Joe had a 32 ft head start over the mosquito)

Coming back, Albert again flies at 5 fps but this time
for only 4 seconds, so it reaches the green paint can 20 feet
from the roller, which is the answer to Q.4)

So, as Sam sees it, Albert takes 16 seconds to reach Joe, flying at
5-3 = 2 fps, and 4 seconds to return, flying along the ladder at
5+3 = 8 fps.

Now we think like Einstein with his mosquito brain. Sam wants to know
when the mosquito reached Joe.

He isn't able to see the mosquito, its too small at 32 feet away,
so he guesses that since it went 32 ft each way, and took 20 seconds to
fly
away and back again, it must have reached Joe after 10 seconds = 1/2 of
20.

So we explain it carefully. First we label the red paint can "A" and the
dropped roller "B".  We write:

If at the point A of space there is a clock, an observer called Sam at
the
red paint can determine the time values of events in the immediate
proximity of the red paint can by finding the positions of the hands
which
are simultaneous with these events. If there is at the point B of space
another clock in all respects resembling the one at the red paint can,
it
is possible for an observer Joe at the dropped roller to determine the
time
values of events in the immediate neighbourhood of the roller at B. But
it
is not possible without further assumption to compare, in respect of
time,
an event at the A with an event at the dropped roller. We have so far
defined only an "A time" and a "B time." We have not defined a common
"time" for the red paint can and the dropped roller, for the latter
cannot
be defined at all unless we establish by definition that the "time"
required by a mosquito to travel from the red paint can to the dropped
roller equals the "time" it requires to travel from B to the red paint
can,
A.

Now, we want to do this algebraically, because tomorrow Joe and Sam
might
be carrying a different length of ladder, and we want a general
solution.

So we write:
If we place x'=x-vt, it is clear that a point at rest in the system
ladder
must have a system of values x', y, z, independent of time.

What that means is the ladder's length is x', so that 32  =  80 - 3 *
16,
and doesn't change as time passes. Did you think it would? Well, we'll
have
to see.  Maybe if we water it, it might grow.

According to Einstein, we are to assume the speed of the mosquito is
independent of the speed of Sam (which is fair enough) and also we are
to
assume that the time for the mosquito to make the round trip (20
seconds)
when divided by 2 is equal to the time it took to reach Joe, 16 seconds.

We don't know yet about the 16 seconds, we can only write it
algebraically
and pretend it is 10 seconds.
It is actually written as  x'/(c-v) [or 32/(5-3) in real numbers].

Now we say:

From the origin of system ladder let a mosquito be emitted at the time
tau0
along the ladder to x' (the other end of the ladder), and at the time
tau1
be reflected thence (that just means go back) to the origin of the
co-ordinates (which we are deliberately vague about as to whether we
mean Sam on the ladder or the red paint can), arriving there at the time
tau2; we then
must have (don't you just love that phrase, "then must have" ?)

½(tau0 + tau2) = tau1,
or ½([midmorning + 0] + [midmorning + 20]) = [midmorning + 16], which is
curious to say the least, since Sam and Joe could be doing this in the
late afternoon for all the difference it would make.
But ok, Einstein wanted to be complete, so I guess its fine.

But our hero and physics wizard isn't satisfied with this. Oh no, we
need
to include the length of the ladder as well, or we won't have any
spacetime
to prattle on about later so that people will see just how smart we are.

Here is Einstein's equation:
½[tau(0,0,0,t)+tau(0,0,0,t+x'/(c-v)+x'/(c+v))] = tau(x',0,0,t+x'/(c-v))

You can read about it at
   http://www.fourmilab.ch/etexts/einstein/specrel/www/
 (in Section 3)

Putting in the mosquito numbers,

½[tau(0,0,0,t)+tau(0,0,0,t+32/(5-3)+32/(5+3))] = tau(32,0,0,t+32/(5-3))
½[tau(0,0,0,t)+tau(0,0,0,t+20)] = tau(32,0,0,t+16)

In agreement with experience (gotta love that phrase!) clearly(!)
(0,0,0,t)
is pretty meaningless, and we can drop the "t+" since we really don't
care
if Sam and Joe are walking in the midmorning or late afternoon.

So,
½ * tau(0,0,0,20) = tau(32,0,0,16).

There's some differentiation by Einstein to make himself look smart and
important, he has to show off all his skills if not his common sense,
because "common sense is the collection of prejudices acquired by age
eighteen", he tells us, and he eventually arrives at

tau  = (t-vx/c^2) / sqrt( 1 - v^2/c^2 )
xi   = (x-vt)     / sqrt( 1 - v^2/c^2 )
eta  = y
zeta = z.

That is what you get when you treat time as if it were a vector and mix
in
some distance.
We can forget y and z, the mosquito didn't fly up into a tree or into
the
ditch at the side.

We apply this to the equations derived:

tau = (16 - 3 * 80 / 25)  /  sqrt (1 - 3^2/5^2)
   = (6.4) / 0.8
   = 8 seconds

xi = 32 / sqrt (1 - 9 / 25)
  = 40 feet

Sanity check:

c = 40 ft / 8 seconds  = 5 fps. Yep, that's the right speed for Albert.

So...
We are standing at the roadside watching Sam and Joe carry a 40 ft
ladder
that they think is a 32 ft ladder, because the speed of mosquitoes is 5
fps
in all inertial frames of reference.

It must be right, its only algebra after all is said and done.

So now you should be able to fully understand Special Relativity, all
you need do is replace the speed of the mosquito with the speed of
light,
have Sam and Joe run at the relativistic speed of 0.6c, the algebra is
perfect, and who needs common sense anyway?

Just remember that 40 ft ladders shrink to 32 ft ladders when you run
with
them at 180,000 km/sec, and you'll be as smart as Einstein the cretin.

For myself, I prefer to keep the collection of prejudices I acquired by
the time I was eighteen.

Androcles