relativistic mechanics relativistic mass and momentum

Relativistic Mechanics

Relativistic Mass and Momentum


In classical mechanics, momentum is always conserved.


In classical mechanics, momentum is always conserved.

However this is not true under a Lorentz transform.


If momentum is conserved for an observer in a reference frame S


If momentum is conserved for an observer in a reference frame , it is not conserved for another observer in another inertial frame .

S

S

Relativistic Mass

After

V=0

Stationary Frame

Before

v

1

v

2

Relativistic Mass

Moving Frame of mass 1

Before

1

v′x

2

After

V′

Relativistic Mass

In the stationary frame,

However in the moving frame, P final > P initial.

Pi = mv – mv = 0 (assume the masses are equal

Relativistic Momentum

For momentum to be conserved the expression for the momentum must be rewritten as follows:

22

0

1 cv

vmP

(Relativistic Momentum)

Relativistic Mass

Where is the rest mass.0m

Relativistic Mass

The expression is produced by assuming that mass is not absolute.

Instead the mass of an object varies with velocity i.e. . vm

Relativistic Mass

The expression for the relativistic mass is

22

0

1 cv

mm

(Relativistic mass)

Relativistic Mass

From the expression for the momentum, the relativistic force is defined as,

dt

PdF

Stationary Frame

Pi = mv – mv = 0 (assume the masses are equal)

Pf = (m1+m2)x0 = 0

Pi = Pf . Therefore Momentum is conserved.

After

V=0

Before

v

1

v

2

Moving Frame of mass1

22

22

21

11

11c

vuvu

m

c

vuvu

mPx

x

x

xi

Before

1

v′x

2

After

V′


2222

22

21

11 )(

1111cvvvv

m

cvvvv

m

cvuvu

m

cvuvu

mPx

x

x

xi

Before

1

v′x

2

After

V′


22

2222

22

21

11

1

2

)(1111

cv

mvcvvvv

m

cvvvv

m

c

vuvu

m

c

vuvu

mPx

x

x

xi

Before

1

v′x

2

After

V′


Before

1

v′x

2

After

V′

212

cvVvV

mPx

xf


Before

1

v′x

2

After

V′

22

01

02

12

cvv

m

c

vVvV

mPx

xf


Before

1

v′x

2

After

V′

mv

cvv

m

cvVvV

mPx

xf 2

01

02

12

22


Hence momentum is not conserved.


Now consider what happens when the modification to the mass is introduced.


Recall the equations for the final and initial momentum.

222 1

2

cv

mvvmPi

vMVMVmmPf )( 21

where 222 1

2

cv

vv

where vV


Consider the initial momentum

m

221

2

cv

mvPi

where 2mreally corresponds to the mass

NB: 01 mm Since it is at rest in the moving frame


Consider the initial momentum

221

2

cv

mvPi

2

22

0

1cv

mm

(Relativistic mass)


222 1

2

cv

vv

2

22

0

1cv

mm

where


222 1

2

cv

vv

2

22

0

1cv

mm

2

222

0

121

1

cvv

c

mm

where


222 1

2

cv

vv

2

22

0

1cv

mm

222

22

0

2

222

0

)1()2(

11

211 cv

cv

m

cvv

c

mm

where


222 1

2

cv

vv

2

22

0

1cv

mm

222

22

0

2

222

0

)1()2(

11

211 cv

cv

m

cvv

c

mm

where

222

222

0

)1()1(

cvcv

mm

simplify


222 1

2

cv

vv

2

22

0

1cv

mm

222

22

0

2

222

0

)1()2(

11

211 cv

cv

m

cvv

c

mm

where

)1(

)1(

)1()1(

22

22

0

222

222

0

cv

cvm

cvcv

mm

simplify


Therefore the initial momentum

221

2

cv

mvPi



)1(

)1(

1

2

1

222

220

2222 cv

cvm

cv

v

cv

mvPi



)1(

)1(

1

2

1

222

220

2222 cv

cvm

cv

v

cv

mvPi

220

1

2

cv

vmPi


Now consider the final momentum

vMVMVmmPf )( 21

2

20

2

2

0

1

2

1cv

m

cv

MM

220

1

2

cv

vmPf


Hence momentum is conserved.


Consider the difference in the relativistic momentum and classical momentum.

Relativistic Energy

Relativistic Energy

To derive an expression for the relativistic energy we start with the work-energy theorem

Relativistic Energy

To derive an expression for the relativistic energy we start with the work-energy theorem

dxdt

dPW

W K Fdx (Work done)

Relativistic Energy

Using the chain rule repeatedly this can rewritten as

vdvdv

dPW

Relativistic Energy


vdvdv

dPW

v

vdvcv

vm

dv

d

022

0

1

Relativistic Energy


vdvdv

dPW

v

vdvcv

vm

dv

d

022

0

1

(Work done accelerating an object from rest some velocity)

Relativistic Energy

v

vdvcv

vm

dv

d

022

0

1

v

dvcv

vmW

022

0

23

1

Relativistic Energy

v

vdvcv

vm

dv

d

022

0

1

v

dvcv

vmW

022

0

23

1 dv

cv

vm

2

322

01

Relativistic Energy

v

vdvcv

vm

dv

d

022

0

1

v

dvcv

vmW

022

0

23

1 dv

cv

vm

2

322

01

Integrating by substitution:

Relativistic Energy

v

vdvcv

vm

dv

d

022

0

1

v

dvcv

vmW

022

0

23

1 dv

cv

vm

2

322

01

Integrating by substitution:

v

dudv

vdvduvuLet

2

22

Relativistic Energy

v

du

cu

vmW

21 23

20

Relativistic Energy

v

du

cu

vmW

21 23

20

23

2

0

12 cu

dum

Relativistic Energy

v

du

cu

vmW

21 23

20

23

2

0

12 cu

dum

u

c

cum

02

20

21

1

2

21

Relativistic Energy

v

du

cu

vmW

21 23

20

23

2

0

12 cu

dum

u

c

cum

02

20

21

1

2

21

u

cucm0

220

21

1

Relativistic Energy

20

220

21

1 cmcucmW

Relativistic Energy

20

220

21

1 cmcucmW

Therefore

2

022

20

21

1cm

cv

cmW

Relativistic Energy

20

220

21

1 cmcucmW

Therefore

2

022

20

21

1cm

cv

cmW

However this equal to K

Relativistic Energy

20

220

21

1 cmcucmW

Therefore

2

022

20

21

1cm

cv

cmW

However this equal to K

therefore the Relativistic Kinetic energy is

2

022

20

21

1cm

cv

cmK

Relativistic Energy

The velocity independent term ( ) is the rest energy – the energy an object contains when it is at rest.

20cm

Relativistic Energy (Correspondence Principle)

At low speeds we can write the kinetic energy as

cv

11 21

2220

cvcmK



cv

11 21

2220

cvcmK

Using a Taylor expansion we get,



cv

11 21

2220

cvcmK

Using a Taylor expansion we get,

1...1322

165222

8322

212

0 cvcvcvcmK


Taking the first 2 terms of the expansion



11 22212

0 cvcmK



11 22212

0 cvcmK

So that

202

1 vmK (classical expression for the Kinetic energy)

Relativistic Energy

Note that even an infinite amount of energy is not enough to achieve c.

Relativistic Energy

The expression for the relativistic kinetic energy is often written as

cmKcv

cm022

20

1

Relativistic Energy


cmKcv

cm022

20

1

or equivalent as 20

2 cmKmc

Relativistic Energy


cmKcv

cm022

20

1

or equivalent as 20

2 cmKmc

where 2mcE is the total relativistic energy

Relativistic Energy

So that 20cmKE

Relativistic Energy

So that

If the object also has potential energy it can be shown that

20cmKE

20

2 cmVKmc

Relativistic Energy

The relationship is just Einstein’s mass-energy equivalence equation, which shows that mass is a form of energy.

2mcE

Relativistic Energy

An important fact of this relationship is that the relativistic mass is a direct measure of the total energy content of an object.

Relativistic Energy


It shows that a small mass corresponds to an enormous amount of energy.

Relativistic Energy


It shows that a small mass corresponds to an enormous amount of energy. This is the foundation of nuclear physics.

Relativistic Energy

Anything that increases the energy in an object will increase its relativistic mass.

Relativistic Energy

In certain cases the momentum or energy is known instead of the speed. The relationship involving the momentum is derived as follows:

Relativistic Energy

In certain cases the momentum or energy is known instead of the speed.

Relativistic Energy

Note that 22

420422

1 cv

cmcmE

Relativistic Energy

Note that 22

420422

1 cv

cmcmE

420

2242 1 cmcvcm

Relativistic Energy

420

2242 1 cmcvcm

Note that 22

420422

1 cv

cmcmE

420

222242 cmvcmEcm

Relativistic Energy

420

2242 1 cmcvcm

Note that 22

420422

1 cv

cmcmE

420

222242 cmvcmEcm

Since mvP

Relativistic Energy

Note that 22

420422

1 cv

cmcmE

420

2242 1 cmcvcm

420

222242 cmvcmEcm

Since mvP 42

0222 cmcPE

Relativistic Energy

When the object is at rest, so that 0p2mcE

Relativistic Energy

When the object is at rest, so that

For particles having zero mass (photons) we see that the expected relationship relating energy and momentum for photons and neutrinos which travel at the speed of light.

0p2mcE

pcE

relativistic mechanics relativistic mass and momentum

Documents