Relativistic Mechanics Relativistic Mass and Momentum.
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Transcript of Relativistic Mechanics Relativistic Mass and Momentum.
Relativistic Mechanics
Relativistic Mass and Momentum
Relativistic Mechanics
In classical mechanics, momentum is always conserved.
Relativistic Mechanics
In classical mechanics, momentum is always conserved.
However this is not true under a Lorentz transform.
Relativistic Mechanics
If momentum is conserved for an observer in a reference frame S
Relativistic Mechanics
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
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′
Moving Frame of mass1
2222
22
21
11 )(
1111cvvvv
m
cvvvv
m
cvuvu
m
cvuvu
mPx
x
x
xi
Before
1
v′x
2
After
V′
Moving Frame of mass1
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′
Moving Frame of mass1
Before
1
v′x
2
After
V′
212
cvVvV
mPx
xf
Moving Frame of mass1
Before
1
v′x
2
After
V′
22
01
02
12
cvv
m
c
vVvV
mPx
xf
Moving Frame of mass1
Before
1
v′x
2
After
V′
mv
cvv
m
cvVvV
mPx
xf 2
01
02
12
22
Moving Frame of mass1
Hence momentum is not conserved.
Moving Frame of mass1
Now consider what happens when the modification to the mass is introduced.
Moving Frame of mass1
Recall the equations for the final and initial momentum.
222 1
2
cv
mvvmPi
vMVMVmmPf )( 21
where 222 1
2
cv
vv
where vV
Moving Frame of mass1
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
Moving Frame of mass1
Consider the initial momentum
221
2
cv
mvPi
2
22
0
1cv
mm
(Relativistic mass)
Moving Frame of mass1
222 1
2
cv
vv
2
22
0
1cv
mm
where
Moving Frame of mass1
222 1
2
cv
vv
2
22
0
1cv
mm
2
222
0
121
1
cvv
c
mm
where
Moving Frame of mass1
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
Moving Frame of mass1
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
Moving Frame of mass1
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
Moving Frame of mass1
Therefore the initial momentum
221
2
cv
mvPi
Moving Frame of mass1
Therefore the initial momentum
)1(
)1(
1
2
1
222
220
2222 cv
cvm
cv
v
cv
mvPi
Moving Frame of mass1
Therefore the initial momentum
)1(
)1(
1
2
1
222
220
2222 cv
cvm
cv
v
cv
mvPi
220
1
2
cv
vmPi
Moving Frame of mass1
Now consider the final momentum
vMVMVmmPf )( 21
2
20
2
2
0
1
2
1cv
m
cv
MM
220
1
2
cv
vmPf
Relativistic Momentum
Hence momentum is conserved.
Relativistic Momentum
Consider the difference in the relativistic momentum and classical momentum.
Relativistic 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
Using the chain rule repeatedly this can rewritten as
vdvdv
dPW
v
vdvcv
vm
dv
d
022
0
1
Relativistic Energy
Using the chain rule repeatedly this can rewritten as
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
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
Relativistic Energy (Correspondence Principle)
At low speeds we can write the kinetic energy as
cv
11 21
2220
cvcmK
Using a Taylor expansion we get,
Relativistic Energy (Correspondence Principle)
At low speeds we can write the kinetic energy as
cv
11 21
2220
cvcmK
Using a Taylor expansion we get,
1...1322
165222
8322
212
0 cvcvcvcmK
Relativistic Energy (Correspondence Principle)
Taking the first 2 terms of the expansion
Relativistic Energy (Correspondence Principle)
Taking the first 2 terms of the expansion
11 22212
0 cvcmK
Relativistic Energy (Correspondence Principle)
Taking the first 2 terms of the expansion
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
The expression for the relativistic kinetic energy is often written as
cmKcv
cm022
20
1
or equivalent as 20
2 cmKmc
Relativistic Energy
The expression for the relativistic kinetic energy is often written as
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
An important fact of this relationship is that the relativistic mass is a direct measure of the total energy content of an object.
It shows that a small mass corresponds to an enormous amount of energy.
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.
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