3063 Exam1 Solutions Sp07

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PHY3063 Spring 2007 R. D. Field

Exam 1 Solutions Page 1 of 6 February 1, 2007

PHY 3063 Exam 1 Solutions

Problem 1 (20 points):Circle true or false for the following 5 questions (1 point each):

(a) (True or False,) The laws of physics are invariant under a change in inertial frame.

(b) (True or False,) The speed of light in a vacuum is independent of the velocity of the sourceof the light, but depends on the velocity of the observer.

(c) (True or False, The mass and charge of a particle depends on its speed.

(d) (True or False,) Light has no rest mass, but it has momentum and energy.

(e) (True or False) The energy of a photon increases with increasing wavelength.

Write in the answer to the following 5 questions (3 points each):

(f) What is the speed = v/c of a proton with momentum p = 1 GeV/c?

Answer: = 0.729Solution:We know that

729.0)GeV93828.0()1(

1

)()(22222

+=

+==

GeV

GeV

cmpc

pc

E

pc

p

(g) What is the Kinetic Energy of a proton with momentum p = 1 GeV/c?

Answer: (in MeV) = 432.44 MeVSolution:We know that

MeVMeVcmKE p 44.432)28.938)(1461.1()1(2 == .

(h) What is the speed = v/c at which the kinetic energy of a particle (with rest mass m0) isequal to its rest mass energy?

Answer: = 0.866Solution:Setting the kinetic energy equal to the rest mass energy gives

2

0

2

0)1( cmRMEcmKE === .

Thus, 11= , and = 2 and

866.02

3

4

3

4

11

11

2====

.

(i) What is the speed = v/c at which a the relativistic mass of an electron is equal to the rest

mass of a 0

meson?

Answer: = 0.9999928Solution:We see that

22 cmcm oe = and 1096.264511.0

134.962

2

==cm

cm

e

o .

Hence,

9999928.01

12=

.


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(j) What is the wavelength of light (i.e. a photon) which has a relativistic mass equal to the rest

mass of an electron?

Answer: (in fm) = 2,426.32 fmSolution:The relativist mass of an electron is given by m = E/c

2and hence

fmMeV

fmMeV

cm

hc

E

hc

pc

hc

p

h

e

32.426,2511.0

85.239,12

===== .

Problem 2 (40 points)Suppose Jim, John, and Rick were all born simultaneously in

the Earth frame (i.e.O-frame). Jim and Rick were born at

the origin at t = 0 and John was born on planet Zilch 49 light-years from Earth (as measured in the O-frame). Rick is put in

a rocket ship at the moment of birth heading for John at speed

= V/c = 0.98. Let the O'-frame be the rocket frame (i.e.

Ricks frame) and 1 Ly = 1 light-year.

Part A (3 points): How long does it take Rick to get to John

according to Jim (i.e. the O-frame)? (Express your answer in

years and round off to the nearest year.)

Answer: 50 years

Solution:Distance is equal to velocity times time in every frame. Hence

yearsyearsyearscLycL

V

Lt 50

98.0

4949/)49(/======

.

Part B (3 points): How long does it take Rick to get to John according to Rick (i.e. the O'-frame)? (Express your answer in years and round off to the nearest year. )

Answer: 10 years

Solution:We see that

yearsyearst

t 1095.90252.5

50' ===

.

Part C (12 points): According to Jim on earth (i.e. the O-frame) at Ricks meeting with John:Ricks age (in years rounded off to the nearest year) =

Jims age (in years rounded off to the nearest year) =

Johns age (in years rounded off to the nearest year) =Ricks rocket traveled a distance (in Ly) =

Jim (i.e. the Earth) traveled a distance (in Ly) =

John (i.e. planet Zilch) traveled a distance (in Ly) =Answers:

Ricks age (in years rounded off to the nearest year) = 50 years

Jims age (in years rounded off to the nearest year) = 50 years

Johns age (in years rounded off to the nearest year) = 50 yearsRicks rocket traveled a distance (in Ly) = 49 Ly

Jim (i.e. the Earth) traveled a distance (in Ly) = 0

John (i.e. planet Zilch) traveled a distance (in Ly) = 0

y

x

z

y'

z'

x'

V= 0.98c

O O'

x = 49 Ly

RickJohn

Jim


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Part D (12 points): According to Rick on the Rocket (i.e. the O-frame) at his meeting with

John:Ricks age (in years rounded off to the nearest year) =

Jims age (in years rounded off to the nearest year) =

Johns age (in years rounded off to the nearest year) =

Ricks rocket traveled a distance (in Ly) =Jim (i.e. the Earth) traveled a distance (in Ly) =

John (i.e. planet Zilch) traveled a distance (in Ly) =

Answers:

Ricks age (in years rounded off to the nearest year) = 10 Years

Jims age (in years rounded off to the nearest year) = 251 years

Johns age (in years rounded off to the nearest year) = 241Ly + 10Ly = 251 yearsRicks rocket traveled a distance (in Ly) = 9.75 Ly

Jim (i.e. the Earth) traveled a distance (in Ly) = 246 Ly

John (i.e. planet Zilch) traveled a distance (in Ly) = 246 Ly

Solution: Distance is equal to velocity times time in every frame. Hence, Ricks believes his

rocket traveled a distance d' given by LyLyctVtd 75.9)95.9(98.0''' ==== .

The four events defined in Part E (in the O-frame) are as follows:

O-frame

A = (0,0)B = (0, 49Ly)

C = (50 Ly, 49 Ly)

D = (50 Ly, 0)with P = (ct, x). Transforming event A to the O'-frame yields A' = (0,0). Transforming event B

to the O'-frame yields

LyLyxctxx

LyLyxxctct

BBBB

BBBB

246)49(0252.5)('

241)49(9247.4)('

===

===

Transforming event C to the O'-frame yields

0))50(98.049(0252.5)('

10))49(98.050(0252.5)('

===

==

LyLyctxx

LyLyLyxctct

CCC

CCC

Transforming event D to the O'-frame yields

LyLyctctxx

LyLyctxctct

CCCC

DDDD

246)50(9247.4)('

251)50(0252.5)('

===

===

Hence,

O'-frame

A' = (0,0)

B' = (-241Ly, 246Ly)C'= (10 Ly, 0)

D'= (251Ly, -246Ly)


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Define the following four events:

Event Description

A Jim and Rick born on Earth at x = 0 at t=0.

B John born on Zilch x = 49 ly at t =0.

C Rick and John meet on the planet Zilch.D Jim when Rick meets John.

Part E (10 points): Plot the four events A, B, C, and D on the space-time plots below and labelthe (ct, x) coordinates of each event in the O-frame and the (ct',x') coordinates of each event in

the O'-frame. (Express the coordinates in terms of light-years, Ly).

.

A B

Frame O

x

ct

ct=50 ly

x=49 ly

CD

Rick

A

C

Frame O'

x'

ct'

John

B

D

Jim

Rick

Problem 3 (40 points)A

0meson decays at rest into two photons in the O-frame (i.e.

0 + ). In the O'-frame the two photons are back-to-back

(' = 180o) and are traveling along the y'-axis (i.e. px = 0 forboth photons, case 1). In the O'-frame photon 1 is traveling up

(i.e. py' > 0) and photon 2 is traveling down (i.e. py' < 0). The O'-frame is moving to the right along the x-axis also with speed Vrelative to the O-frame (the origins of the two frames coincide at t

= t' = 0 and c = 3x108

m/s).

Part A (3 points): What is the energy and wavelength of

photon 1 in the rest frame of the 0

(i.e. O'-frame)? (Express youanswers in MeV and fm = 10

-15m, respectively)

Answers:

E'1 (in MeV) = 67.48 MeV

'1 (in fm) = 18.37 fm

Solution:In the rest frame of the 0

momentum conservationyields

2100 ppprrr

+== and hence |||| 21 ppprr

== .

Thus,

y

x

z

y

z

x

VO'O

0 meson

0 meson at rein the O'-fram

Before Decay

y

z

x

O'

Two back-to-back

photons in the O'-frame

After Decay (case 1)


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121

2 200 EEEcME =+== and hence MeVMeVcMEE 48.67)96.134(212

21

21

1 00 === .

For photons

fmMeV

fmMeV

E

hc

pc

hc

p

h37.18

48.67

85.239,1

==== .

Part B (3 points): What is the energy and wavelength of photon 2 in the rest frame of the 0

(i.e. O'-frame)? (Express you answers in MeV and fm = 10-15

m, respectively)

Answers:

E'2 (in MeV) = 67.48 MeV

'2 (in fm) = 18.37 fm

Solution:In the rest frame of the 0

the two photon have the same (magnitude) momentum andhence the same energy and wavelength.

Part C (12 points): If an observer in the O-frame observes the angle between the two photons

to be = 90o, what is the speed = V/c of the O'-framerelative to the O-frame and what is the energy and wavelength

of the two photons (as observed in the O-frame)? (Express youanswers in MeV and fm = 10

-15m)

Answer:

707.02/1/ == cV

E1 (in MeV) = 95.43 MeV

1 (in fm) = 12.99 fm

E2 (in MeV) = 95.43 MeV

2 (in fm) = 12.99 fm

Solution:In the rest frame of the O'-frame photon 1 has velocity p'x = 0 and cp'y = E'1. Usingthe Lorentz transformations gives

1111

')')'(()(

EEcpcpxx

=+= and111

')'()(

Ecpcpyy

== .

Hence

1

)(

)(tan

1

1

1 ==x

y

p

p.

Now we square both sides and solve for,

1

22

22

tan

1

1)(

=

= and hence 707.0

2

1

11

1

tan1

1

1

2=

+=

+=

where I used 1)45tan(tan 1 ==o

. Note that if the angle between the two photon is 90o, then

photon 1 makes an angle of 45o

and photon 2 makes an angle of -45o

with the x-axis. I will use

the Lorentz transformations to calculate Eg1 as followsMeVMeVEcpEE x 43.95)48.67(2')''( 111 ==+=

where I uses the fact that p'x(1) = 0 and 21/12 == . Photon 2 has the same energy as

photon 1. Also,

fmMeV

fmMeV

E

hc99.12

43.95

85.239,1

== .

y

z

O

Two photons inthe O-frame


= 90ox


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Part D (12 points): Suppose instead that in the O'-frame the two photons are back-to-back ('= 180

o) and are traveling along the x'-axis (i.e. py' = 0 for both

photons, case 2). In the O'-frame photon 1 is traveling to the right

(i.e. px' > 0) and photon 2 is traveling to the left ( i.e. px' < 0). If theO'-frame is traveling at the same speed as in part C, what is the

energy and wavelength of the two photons (as observed in the O-frame)? (Express you answers in MeV and fm = 10-15

m)

Answer:

E1 (in MeV) = 162.91 MeV

1 (in fm) = 7.61 fm

E2 (in MeV) = 27.95 MeV

2 (in fm) = 44.36 fm

Solution:I will use the Lorentz transformations to calculate E1 as follows

MeVMeVEcpEE x 91.162)48.67(4142.2')1()''( 111 =+=+=

where I used cp'x(1) = E'1. Thus,

fmMeV

fmMeV

E

hc61.791.162

85.239,1

1

1

==

.

For photon 2 we have

MeVMeVEcpEE x 95.27)48.67(4142.0')1()''( 222 ==+=

where I used cp'x(2) = -E'2. Thus,

fmMeV

fmMeV

E

hc36.44

95.27

85.239,1

2

2

==

.

Part E (10 points): Show that the energy is conserved in the O-frame for both case 1 and case

2 by showing explicitly that E0 = E1 + E2 in both cases.

Solution:For case 1 we have

22111 ' cMEE o == and 22122 ' cMEE o == .

Thus,

oo EcMEE ==+2

21 .

For case 2 we have2

21

11 )1(')1( cMEE o +=+= and2

21

22 )1(')1( cMEE o == .

Thus,

oo EcMEE ==+2

21 .

y

z

x

O'

Two back-to-back

photons in the O'-frame


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3063 Exam1 Solutions Sp07