phys 342 - lecture 7 notes - f12

8/11/2019 PHYS 342 - Lecture 7 Notes - F12

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Lecture 7

Relativistic Dynamics

In Newtonian dynamics, the mass of an object is assumed

to be constant, independent of how fast the object moves.

Consequently, Newtons second lawF = ma

implies that an object can acquire a constant

acceleration when acted upon by a constant force

and it should thus have no problem of achieving

a velocity greater than the speed of light,

a violation of one of theEinsteins relativistic

principles.


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Lecture 7

Experimental Results


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Lecture 7

Dependence of Mass on Velocity

Two identical particles move at the same velocity, as measured

in a laboratory frame S, but along opposite directions. The two

particles collide head-on inelastically.

u u

x

S

Before the collision:

x

SS

x

uS

x

After the collision:

M0m(u)m(u)


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Lecture 7

Two Views of an Inelastic Collision

In S, we have uvuux

=!= , , so, in S, the velocity of the

other particle before collision is

U

cuu

cu

uu

c

vuvuu

x

x

x !"

+

!=

+

!!=

!

!=#

2

2

2

2

2 1

2

11

and the velocity of the large particle after collision is u!

Momentum conservation: ))(())(( uuMUUm !=!

Mass conservation: )()(0

uMmUm =+


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uU

u

m

Um

!

=

0

)(

EliminatingM(u)from the equations, we have

On the other hand, we have

U(1+ u2

c2) =2u!u2 c2 +1=2u U

u2 " 2 c2 U( )u + c2 =0

#u = c

2

U1 1"

U2

c2

$

%&&

'

())

Lecture 7

Two Views of an Inelastic Collision-Contd


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Lecture 7

Two Views of an Inelastic Collision-Contd

Since we must have u U/2for U


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Lecture 7

Relativistic Mass Formula

( ) )(

1

1)(2122

0

UcUm

Um!"

#

=

In non-relativistic regime (U


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Lecture 7

Constant Force Revisited

A constant force,F, acts on an object of rest mass m0. Illustrate

that the velocity of the object can never exceed the speed of light.

220

22

0222

0

2

0

22

000

00

00

1

,

1),(

)(),(

cmtF

mFtuuu

cm

Ft

m

Ft

cu

umumFtumdFdt

umdFdtum

dt

d

dt

dpF

uu

ut

uu

+

==!"

#$%

&'!

"

#$%

&

'==(=(

===

))

))

cut !"! ,


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Lecture 7

Work and Energy

Work done by a forceFon an object in distance dxis given

by dW = F dx.

In Newtonian mechanics, we have

TvmvdvmW

vdvmvdtdt

dvmFdxdW

v

!="=

=#$%&

'(

==

20

00

00

2

1

)(

In relativistic mechanics, we have

)()()(

00

vvdmvdtdt

vmdFdxdW v

v!

!="

$%'

==


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Lecture 7

Work and Energy-Contd

v

v

v

c

vcm

cv

vm

dv

cv

vm

cv

vm

cv

vdvmW

0

21

2

22

022

2

0

0 22022

2

0

220

0

1

1

11

1

!!"

#

$$%

&''(

)**+

,-+

-=

.-

--

=

''

(

)**

+

,

-/.=


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Lecture 7

Work and Energy-Contd

2

02

2

2

2

22

2

0

2

0

21

2

22

022

2

0

1

1

1

1

cm

c

v

c

v

cv

cm

cm

c

vcm

cv

vmW

!"#

$%&

'!+

!=

!(()

*++,

-!+

!=

)1(20 !

= vcmW "

Kinetic energy:

WcmcmcmET v =!=!"2

0

2

0

2

0 #


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Lecture 7

phys 342 - lecture 7 notes - f12

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