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The Beauty of Math

The Beauty of Math - Physics Forum

The Beauty of Math - Physics Forum. Discuss and ask physics questions, kinematics and other physics problems.

#11
01-31-2005, 04:37 PM
 Androcles Guest Posts: n/a
The Beauty of Math

"PD" <[Only registered users see links. ]> wrote in message

Nor will you. Nor will you.
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 named 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?

..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
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.

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 = ½ 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 will 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 A with an event at the dropped roller, B. 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.

Note the *definition*. Remember this is hypothetical, not real. The
definition is very important.

Now, we want to do this algebraically, because tomorrow Joe and Sam
might
be carrying a different length of ladder, running at a different speed,
whatever, 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
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 Albert, 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,
by Albert's DEFINITION.

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.
It is very important to include the length of the ladder into the
equation.
You'll see why later.

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))

[Only registered users see links. ]
(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).

Now do you see why we need to include the length of the ladder into the
evaluation of time? We cant just say ½ * 20 = 16 without it. Even my
grandson would say that wasn't right, and he's not learning algebra yet.

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", or so 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
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'll keep the collection of prejudices I acquired by the
time I was eighteen.

Androcles

#12
01-31-2005, 04:39 PM
 sammy Guest Posts: n/a
The Beauty of Math

You forgot A= -1

#13
01-31-2005, 05:01 PM
 Greysky Guest Posts: n/a
The Beauty of Math

"Androcles" <[Only registered users see links. ]> wrote in message
news:i9tLd.33022\$[Only registered users see links. ].blueyonder.co.uk ...

[snip]

Ummm... mayhaps your mosquitos' name is Albertina??
It is common knowledge amongst mosquitos that only the females bite and
drink blood, which they use to feed the babies. The male mosquitos, are like
human males, mostly large clumsy fellows who are happy go lucky in their
attitude towards life. They would never bite a mammal for its blood. Because
of this foible, most male mosquitos would rather be mated with females
outside their family line, but are too stupid to know the difference...

#14
01-31-2005, 05:28 PM
 PD Guest Posts: n/a
The Beauty of Math

Androcles wrote:
are
years
I
these
like
Any
like
box
won't
when
fps
the
so.
paint
of
swat

to
1/2 of
the
at
immediate
space
can,
the
But
dropped
paint
speed,

we'll
are
seconds,
time

time
which is
the
are.
tau(x',0,0,t+x'/(c-v))
tau(32,0,0,t+32/(5-3))

the
my
yet.
and
mix

Albert.
5
is

cretin.
I think it's kinda cute the way you follow me around, Androcles...

PD

#15
01-31-2005, 05:39 PM
 Eugene Shubert Guest Posts: n/a
The Beauty of Math

Anthony,

I commend you for wanting to learn the math of physics. For
general relativity, you'll need to learn differential geometry.
Quantum mechanics is based on the language of Hilbert spaces.
For a clear understanding of special relativity, which is basic
to all of modern physics, here's a elegant introduction to that
subject from a mathematician's point of view:
[Only registered users see links. ]

#16
01-31-2005, 06:22 PM
 compredez Guest Posts: n/a
The Beauty of Math

On 31 Jan 2005 07:48:31 -0800, "PD" <[Only registered users see links. ]> wrote:

1. Beauty is in the eyes of the beholder.
2. You appear to be a killjoy.
#17
01-31-2005, 06:30 PM
 Androcles Guest Posts: n/a
The Beauty of Math

"sammy" <[Only registered users see links. ]> wrote in message

A frequency of -1 and a time of -1.... Yeah...gotcha.
Androcles.

#18
01-31-2005, 06:39 PM
 PD Guest Posts: n/a
The Beauty of Math

compredez wrote:
them
Equations,
later,
else
like
(especially
You

1. Agreed. And many mathematicians (and Martin Gardner, RIP) find joy
in mathematics alone. So do I, to a point.
2. Maybe. But also practical. Recall the original poster wanted:

So I offered a realistic path to obtaining that. If his interest and
enthusiasm for the end goal is real, the realistic path will not kill
his joy but in fact enhance it. It's just going to take a lot of work.
PD

#19
01-31-2005, 07:03 PM
 Jim Sprigs Guest Posts: n/a
The Beauty of Math

Anthony wrote:

Books? First rate is:

Courant & Hilbert "Methods of Mathematical Physics", 2 vols, Wiley

Also:

Morse & Feshbach "Methods of Theoretical Physics", 2 vols, McGraw-Hill

Roman "Some Modern Mathematics for Physicists and Other Outsiders", 2
vols, Pergamon

Umm... better mention something _not_ in two volumes:

Boas "Mathematical Methods in the Physical Sciences" Wiley.

But Courant & Hilbert is by far the best.

#20
01-31-2005, 07:12 PM
 Androcles Guest Posts: n/a
The Beauty of Math

"Greysky" <[Only registered users see links. ]> wrote in message
news:nvtLd.22519\$[Only registered users see links. ].prodigy. com...

Maybe Einstein was a fairy, too. I wouldn't know.

Androcles.

 Tags beauty , math

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