the big boing theory

8/9/2019 The Big Boing Theory

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The Big Boin g The ory

(The exam a t t he end of all exams)

by J oh n Mich ae l William s

jmmwill comcast.net

revised from US ENET post ings of March 2001, to ne ws groups s ci .phy sics ,

sci .physics.relativity, sci .astro

Copyrigh t (c) 2008, J ohn Mich ae l Williams

All Rights Re serve d


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2008-11-27 jmw Big Boing Theory v. 1.2 2

0. The P roblem

Note: The nar ra tive her e is based on rea lly fictit ious events!

Two cosmona ut s ha d been living in an obscure compa rt men t of th e Mir Orbiter for

seven year s, ever since Groun d Cont rol ina dvertent ly omitt ed th em from a list of retur n

passengers.

As usu al, th ey cha rged th eir ion self-propulsion u nits from Mir's main batt eries, and

drew a lit tle fluid from t he a ir condit ionin g unit for imp ulse.

They then jett ed away int o space, far from an y near by object, ta king with t hem a

lar ge, compr essed, coiled sprin g an d a long bu ngee cord, ea ch of finite, bu t van ishingly

small, mass.

One cosmonau t, with all equipment, weighed half as m uch as th e oth er. They each

tied t hem selves t o one en d of th e bun gee cord an d t ook u p positions on opposite en ds of

th e spr ing. They synchr onized th eir watches to th e Big Ivan clock on th e Mir steeple;

an d, exactly at 12:00, released t he lat ch on t he spr ing.

The spr ing expan ded in n egligible time, sendin g th e two cosmona ut s off in exactly

opposite directions.

The sma ller cosmonaut determined by spectr al shift of the fixed sta rs t ha t h er speed

was 0.9c. After a n h our ha d lapsed by her watch, the bun gee cord's slack was u sed up,

an d t he cord ra pidly decelera ted h er a nd pu lled her back in t he opposite direction.

The two cosmona ut s were ret ur ned t o opposite ends of th e spring, compr essing it a nd

resetting the latch.

By the sm aller cosmona ut 's watch, it was exactly 14:00.

Quest ions:

1. What t ime was it by th e lar ger cosmonaut 's watch?

Hint : What t ime would Big Ivan sa y it was?

2. Mir ha ving been deorbitt ed becau se of repeat ed batt ery and a ir conditioning

problems, how will they retur n t o Ea rt h?

3. Should th ey ar gue about the time?

4. F inal Exam :

A. Describe th e stress on the bun gee cord du rin g th e above.B. If you can't describe it, at least explain it.

C. If you can 't explain it, then a t least plea se excuse it.


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I. The First P ass

First of all, this r eally happened!

In 1993, Boris Yeltsin wa s filmed on CNN , walking out on t he t ar ma c to greet t he

cosmonaut s on their retu rn from Mir. He shook their han ds and congra tu lated them,

briefly speak ing to each one individually. As he was gettin g int o his car a fter t heceremony, he was report ed to rema rk (in Russian ),

"Huh . That 's fun ny. I thought ther e were more of th em."

Any way, here is an an swer to Question 1:

The problem is amen able to reasonable additiona l assum ptions. The only error would

be to assign a specific ma ss eith er t o th e sprin g or t he cord--for exa mple, to comput e a

mechanical resonant frequency or something along those lines.

The behavior of th e spring is irr elevan t to the problem. You ma y allocat e it a tin y bit

of momen tu m a nd h ave it s it in th e fram e of Big Ivan; or, let it follow one of th ecosmonaut s or boun ce ar oun d between th em.

Similar ly, you m ay ha ve the bun gee cord sit in th e rest fra me of Big Ivan a nd pa y out

slack t o the cosmonaut s a t both ends; or, you ma y let one of th e cosmonaut s carr y th e

slack an d pay it out to the oth er.

The first problem is very str aight forwa rd: One solut ion would be by defining th e lab

fra me time, tF , as th e time in th e fra me of th e spring before it was r eleased. This would

be th e fra me of Big Ivan . Then , call the proper time of th e light er an d heavier

cosmonauts, tL and tH , respectively.

By Lorentz time dilation,

t t tF L L H H = = , (1)

in wh ich j jv c= 1 12( ) .

So, from (1), th e hea vier cosmona ut 's tim e would be

t tH LL

H

=

. (2)

Incident ally, before cont inu ing, notice th at L was about 1 1 0 81 2 25 . . ; so, th e tFdur at ion of th e excur sion mu st h ave been about 4.5 hour s: It was 16:30 when th ey met

aga in, accordin g to Big Ivan t ime.

They missed Mir, but ma ybe they can cat ch ISS?

To solve Eq. (2), th e quickest wa y would be sim ply to appr oxima te t he h eavier

cosmonaut 's speed by assum ing the doubled mass pu t t ha t speed t wice as far away from c


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as th at of th e light er cosmona ut 's: So, ifv cH 0 8. , then H 1 67. ; and so, from Eq. (2),

we would h ave,

tH = 22 25

1 672 7

.

.. h ou r s, (3)

ma king it a bout 14:40 by the h eavier cosmonaut 's watch.Call th is COARSE ANSWER 1.

More pr ecisely, we know moment um is conser ved an d ma y be set equa lly to 0 for th e

two cosmona ut s in the fra me of th e spring just before release. The relat ivist ic definit ion

of momen tu m of a m ass ive body is,

p m v= ; and so, (4)

m v m vL L L H H H = , (5)

in the fra me of Big Ivan . We need vH . Solving Eq. (5) for H Hv , and using L from ca.

Eq. (2) above, we find t ha t

2 250 9

21

2.

. =

c v

v

c

H

H

. (6)

After some algebra, we find t ha t v cH 0 7. , making H about 1.4. Substitu ting into

Eq. (2) above,

tH = 2 2 25 1 4 3 2( . . ) . hours, (7)

ma king it a bout 15:10 by the h eavier cosmonaut 's watch.

Call this FIN E ANSWER 1.

The quest ion about t he bun gee cord tension will be answer ed soon. Hint : If th e cord is

not to break, wha t would ha ve to be th e speed of soun d in it?


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II. The Secon d P ass

The final question, the stress in the bungee cord, again allows for reasonable

assumptions.

Let's star t by looking at th e length of th e cord . We can't ha ndle a relat ivistic pr oblem

validly by assum ing length an d time t o be independent, but we can calculat e th ema ximu m length possible at an y time, in an y fra me: This is just th e length with no

Loren tz cont ra ction, th e length t o which th e bun gee cord would ha ve to ha ve been

manufactured.

In t he fra me of th e coiled spring just before r elease, we know the t wo cosmona ut s

t ravelled in opposite di rect ions for about 4 5 2 2 25. .= hours.

In t ha t frame, th en, using the previous a nswer, the cord u nwra pped to a length LF

given by their combin ed speed of .9c + .7c = 1.6c. So,

LF=

1 6 3 10 2 25

8

. ( ) . )m s h (60 s h

2

; or, (8)

LF 4 109 km . (9)

This is slight ly less tha n t he avera ge radius of th e orbit of th e planet N eptun e.

Wow! The cord m us t be qu ite st iff, to rever se a velocity of .9c in a t ime n egligible on

th e scale of an h our. It was, of cour se, ma de of na notu bules precipita ted in a face-

cent ered cubic ar ra y of th e newly discovered pa rt icle, the st rongerin o.

Come to thin k of it, I wonder wh at h app ened to the sprin g? It sh ould be stored sa fely

on Mir s omewh ere, wher e some kid couldn 't get h old of it a nd a ccident ally un lat ch it . . ..

Let's assum e for br evity th at th e slack in t he bun gee cord wa s carried by the h eavier

cosmonaut an d paid out to the light er. Looked at in th e fra me of th e lighter cosmonaut ,

th e cord t hen would have to be able to survive being un wrapped a t a speed equa l in tha t

fra me t o wha t we shall call vLH , th e combined s peed of separ at ion.

To add th e two velocities, we ha ve to realize th at th e speeds .9c an d .7c were given in

th e fra me of th e coiled sprin g just before relea se. So, tim e dilat ion will apply to both of

th em: Time passing in t he larger cosmonaut 's fra me will be dilat ed relative to the

dilat ion in th e light er ones. A clock in th e fram e of Big Ivan m erely copying ticks of, say,

th e hea vier cosmona ut 's clock will seem t o run slow by a factor ofH

in th e Big Ivan

fra me. This copy-clock, if viewed from t he fra me of th e light er cosmona ut , will ru n m ore

slowly yet, t his t ime by a factor ofL . Using th e an swers previously calculat ed, th e

gam ma of the u ncoiling point on th e bun gee cord t her efore will be

LH L H= = =2 25 1 4 3 15. . . . Thus, we may write,

3 15 1 1

2

. =

v

c

LH ; an d, (10)


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v cLH 0 95. . (11)

In th e fra me of th e light er cosmonaut , th e cord will be lengthen ing an d propagating a

wave of slight t ension in th e direction of th e hea vier cosmona ut at a speed of0 95. c .

Ther efore, t he speed of sound in th e cord mu st exceed 0 95. c ; oth erwise, th e cord would

separa te a s it was being unwra pped.

As we know, spring keeps ret ur nin g. So, th inkin g again of our lost spr ing . . . while

compr essed, it mu st h ave stored ener gy enough t o accelerat e a combined m ass of, say,

150 kg to 0 95. c . This stored energy may be consider ed equal to th e kinetic ener gy of th e

two separ at ing cosmonau ts.

The r elat ivistic total ener gy is given by E pc m c m c= + =( ) ( )2 2 2 2

. The kinetic

energy EK will be the difference between t he t ota l an d t he m c2 energy residing in m ass;

so,

E m c cK= =

( ) . 1 2 15 150

2 2

3 10

19

jou les. (12)

Sta rt ing at 20 C, it ta kes about 4 540 80 10 3 102 6 10jou le cal cal g g t on + ( ) joules to

vaporize a met ric ton of wat er; so, th e kinet ic ener gy stored in t he compr essed spr ing is

enough to vaporize one billion met ric tons of wat er. Luckily, it is safely out in spa ce,

where nothing ever ha ppens.

But, Mir, ISS, an d everything else actua lly is not in space but r at her in t he th innest

reaches of th e Ea rt h's upper at mosphere! So, th ings can ha ppen.

What h appens when th e slack in th e bun gee cord is all paid out ? An an swer to th is

will be att empted next.


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III. The Third P ass

I'd like to sta rt by apologizing for pa nicking th at way: I thought th e compr essed sprin g

ha d been stolen; and, in a ll the excitement , I misun derstood wha t t hey were saying.

If th ere are a ny members of th e press still present, I'd just like to say that th e thr ee

th ird-grade kids I overhear d t alking about football now have been r eleased int o thecust ody of their pa rent s.

Of cour se, th ey will rema in u nder susp icion of ter rorism for t he r est of th eir lives; but ,

th ink of all the both er of cha nging t he police records .

The bungee cord also clear ly is a problem. We estim at ed its ma ximum length

previously, but t ha t ma y ha ve to be cha nged after considering how it mu st work while

being paid out.

In t he frame of th e lighter cosmonaut , the st ress in t he cord is const an t in t ime during

th e first hour after spring release. Call th is th e slack t ension . However sma ll th e ma ssper un it length of th e cord, as it u nwra ps, the additiona l increment in m ass per u nit t ime

accelerated by the lighter cosmonaut is proportional to the additional length, which is

const an t. In t he fra me of th e heavier cosmonaut , a negligible an d much sma ller const an t

tension ma y be assum ed exert ed in t he direction of the light er one.

Of cour se, during slack t ension, th e tension must be small enough t ha t it does not

noticeably r educe th e combined speed ofv cLH = 0 95. .

When t he slack ha s been consu med, a su dden increa se in ten sion will occur , first felt

by th e heavier cosmonau t, who suddenly has ru n out of cord to pay. Call th is the taut

tension . The taut t ension will propagate as a wave towar d the light er cosmonaut a t th e

speed of sound in th e cord, sa y, 0 98. c .

We might at this point a ssume (a) that th e cord m ust str etch; or, we might disallow

th is and a ccelera te th e heavier cosmonaut (b) imm ediat ely to th e velocity of th e lighter ;

(c) imm ediat ely to th e negat ive of the h eavier velocity; or, perh aps (d) gra dua lly to some

oth er inter media te velocity. The imm ediat e choices ass um e virtu alizat ion of ener gy over

a space-like interval.

We decide here t o use th e first altern at ive, stret ching, becau se we do not wish to

ass um e a specific ma ss for th e cord. We also will assu me immedia te reversa l of th e

velocity of th e hea vier cosmona ut at some unspecified time in the fra me of th e light er. In

reality, the ma ss a nd elast ic const an ts of the cord would determine t he dyna mics of th erest orin g force in all fra mes.

Becau se we know that th e lighter cosmonaut must cont inue at 0 9. c (Big Ivan fram e)

un til the t au t t ension r eaches her , we permit t he cord t o str etch freely for t he dur at ion of

th e propagation of th e tau t t ension wave.

So, using th e reversed frame of th e heavier cosmonaut at th e point of initiat ion of tau t

ten sion, we ma y define an origin of dista ncex by th e locat ion 0 of th e hea vier cosmona ut

at t he insta nt of initiation of th e tau t ten sion wave. Then, in the bungee cord, we ha ve a


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region of ta ut ten sion (T T) in t he fram e of the hea vier cosmonau t pr opagat ing in th e

direction of th e light er cosmona ut an d boun ded by a point xTT obeying,

x c tHTT H= 0 98. , (13)

th e tau t segment of th e cord a t 0 <


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In t he fram e of Big Ivan , the fractional ad ded length of th e cord because of str etching

then would be something like,

stretch = +

4 10 0 47 60 0 992

4 101125

12 2

12

. ..

c. (17)

Thus, a fter adding a str etch of no more th an about 12.5% in t he frame of Big Ivan, th eta ut wave will reach t he light er cosmonaut an d, we assume jerkily, will reverse her

momentum.

With both cosmonaut momenta reversed, a slack wave now will propagate over t he

bungee cord u nt il all tension ha s been relaxed. This concludes th is answer t o th e final

exam quest ion.

This is as far out as I go with th is theory. I hope readers have foun d it hu morous an d

thought-provoking.

the big boing theory

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