Transcript Homework

From last time…

Einstein’s Relativity
◦ All laws of physics identical in inertial ref. frames
◦ Speed of light=c in all inertial ref. frames
• Consequences
– Simultaneity: events simultaneous in one frame
will not be simultaneous in another.
– Time dilation
– Length contraction
– Relativistic invariant: x2-c2t2 is ‘universal’ in that
it is measured to be the same for all observers
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2)
E (m0c3.5
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relativistik
2.5
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non relativistik
1.5
1
0.5
v (c)
0
0
0.5
1
1.5
2
2.5
3
3.5
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Review: Time Dilation and Length Contraction
Time in other
Tp
frame
T  Tp 
2
2
1 v c
Time in object’s rest
frame
Times measured in other
frames are longer
(time dilation)

Length in other
frame
Length in object’s
rest frame

L
Lp

 Lp
2
v
1 2
c
Distances measured in other
frames are shorter
(length contraction)
to define the rest frame
Need
and the “other” frame which is moving with
respect to the rest frame
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Relativistic Addition of Velocities
• As motorcycle velocity
approaches c,
vab also gets closer and
closer to c
• End result: nothing
exceeds the speed of
light
v ad  v db
v ab 
v ad v db
1
2
c
vdb
Frame b
vad
Frame d
Object a
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‘Separation’ between events
Views of the same cube
from two different
angles.
 Distance between
corners (length of red
line drawn on the flat
page) seems to be
different depending on
how we look at it.

• But clearly this is just because we are not considering the
full three-dimensional distance between the points.
• The 3D distance does not change with viewpoint.
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Newton again

Fundamental relations of Newtonian physics
◦
◦
◦
◦
◦

acceleration = (change in velocity)/(change in time)
acceleration = Force / mass
Work = Force x distance
Kinetic Energy = (1/2) (mass) x (velocity)2
Change in Kinetic Energy = net work done
Newton predicts that a constant force gives
◦ Constant acceleration
◦ Velocity proportional to time
◦ Kinetic energy proportional to (velocity)2
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Forces, Work, and Energy in Relativity
What about Newton’s laws?

Relativity dramatically altered our perspective
of space and time
◦ But clearly objects still move,
spaceships are accelerated by thrust,
work is done,
energy is converted.

How do these things work in relativity?
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Applying a constant force

Particle initially at rest,
then subject to a constant force starting at t=0,
momentum = (Force) x (time)
p= F. t
 Using momentum = (mass) x (velocity), (p=mv)
Velocity increases without bound as time increases
Relativity says no.
The effect of the force gets smaller and smaller
as velocity approaches speed of light
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Relativistic speed of particle subject
to constant force


At small velocities
(short times) the
motion is described
by Newtonian physics
At higher velocities,
big deviations!
The velocity never
exceeds the speed of
light
SPEED / SPEED OF LIGHT

Newton
1
Einstein
0.8
0.6
0.4
v

c
0.2
t /to
t /t o   1
2
, to 
F
moc
0
0

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TIME
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
Momentum in Relativity
The relationship between momentum and
force is very simple and fundamental
Momentum is constant for zero force
and
change in momentum
 Force
change in time

This relationship is preserved in relativity
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
Relativistic momentum
Relativity concludes that the Newtonian
definition of momentum
(pNewton=mv=mass x velocity) is accurate at
low velocities, but not at high velocities
Relativistic gamma
The relativistic momentum is:
prelativistic  m v
1

2
1 (v /c)
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mass
velocity
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Was Newton wrong?

Relativity requires a different concept of
momentum
prelativistic  m v
1

2
1 (v /c)

But not really so different!

For small velocities << light speed
1, and so prelativistic  mv


This is Newton’s momentum
Differences only occur at velocities that are a substantial fraction
of the speed of light
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Relativistic Momentum
Momentum can be increased arbitrarily,
but velocity never exceeds c
 We still use
1
change in momentum
 Force
change in time
For constant force we still have
momentum = Force x time,
but the velocity never exceeds c
 Momentum has been redefined

SPEED / SPEED OF LIGHT

Newton’s
momentum
0.8
0.6
0.4
v

c
0.2
p / po
p / po 
2
1
, po  moc
0
prelativistic  m v 
mv

1 (v /c) 2
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2
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RELATIVISTIC MOMENTUM
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Relativistic momentum for
different speeds.
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How can we understand this?
 change in velocity


 change in time 

acceleration
much smaller at high speeds than at low speeds

Newton said force and acceleration related by mass.

We could say that mass increases as speed increases.

prelativistic  mv  mv  mrelativisticv
• Can write this
— mo is the 
rest mass.
prelativistic  mov  mo v  m v

1
1 (v /c)
,
m


m
o
2
— relativistic mass m depends on velocity
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


The the particle
becomes extremely
massive as speed
increases ( m=mo )
The relativistic
momentum has new
form ( p= mov )
Useful way of thinking
of things remembering
the concept of inertia
RELATIVISTIC MASS / REST MASS
Relativistic mass
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4
3
2
1
0
0
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0.2
0.4
0.6
0.8
1
SPEED / SPEED OF LIGHT
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
Example
An object moving at half the speed of light
relative to a particular observer has a rest
mass of 1 kg. What is it’s mass measured by
the observer?
1
1
1



2
2
1 0.25
1 (v /c)
1 (0.5c /c)
1

 1.15
0.75
So measured mass is 1.15kg

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Question
A object of rest mass of 1 kg is moving at
99.5% of the speed of light.
What is it’s measured mass?
A. 10 kg
B. 1.5 kg
C. 0.1 kg
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Relativistic Kinetic Energy

Might expect this to change in relativity.

Can do the same analysis as we did with Newtonian
motion to find
KE relativistic   1moc
 Doesn’t seem to resemble Newton’s result at all
2

However for small velocities, it does reduce to the
Newtonianform (by using power series)
1
KE relativistic  mov 2 for v  c
2
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Relativistic Kinetic Energy


Kinetic energy gets
arbitrarily large as speed
approaches speed of light
Is the same as
Newtonian kinetic
energy for small speeds.
4
2
o
Can see this graphically
as with the other
relativistic quantities
(KINETIC ENERGY) / m c

3
Relativistic
2
1
Newton
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0.2
0.4
0.6
0.8
1
SPEED / SPEED OF LIGHT
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Total Relativistic Energy

The relativistic kinetic energy is
KE relativistic   1moc 2
 moc 2  moc 2
Depends on
velocity
Constant,
independent of
velocity
 this as
• Write
moc 2  KE relativistic  moc 2
Total energy
Kinetic energy
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Rest energy
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
Mass-energy equivalence
This results in Einstein’s famous relation
E  moc , or E  mc
2
2

This says that the total energy of a particle
is related to its mass.

Even when the particle is not moving it has energy.

We could also say that mass is another form of
energy
◦ Just as we talk of chemical energy, gravitational energy,
etc, we can talk of mass energy
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Example

In a frame where the particle is at rest,
its total energy is E = moc2

Just as we can convert electrical energy to
mechanical energy, it is possible to tap mass
energy

How many 100 W light bulbs can be powered
for one year 1 kg mass of energy?
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Example
2

E = moc = (1kg)(3x108m/s)2=9x1016 J of
energy

1 yr=365,25 x 24 x3600s= 31557600 (~30
x106 sec)

We could power n number of 100 W light bulbs
n=E/(P.t)=9x1016/(100x 30 x106 )= 3 x106
(30 million 100 W light bulbs for one year!)
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EXAMPLE
A proton moves at 0.950c. Calculate its (a) the rest
energy, (b) the total energy, and (c) the kinetic
energy!
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EXAMPLE
Determine the energy required to accelerate an
electron from 0.500c to 0.900c
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Energy and momentum
2
 Relativistic energy is
E  moc

Since  depends on velocity, the energy is measured to be different
by different observers

Momentum also different for different observers

◦ Can think of these as analogous to space and time, which individually are
measured to be different by different observers

But there is something that is the same for all observers:
E  c p  m oc
2

2
2

2 2
= Square of rest energy
Compare this to our space-time invariant
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2
x c t
2 2
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A relativistic perspective




The concepts of space, time, momentum, energy
that were useful to us at low speeds for Newtonian
dynamics are a little confusing near light speed
Relativity needs new conceptual quantities,
such as space-time and energy-momentum
Trying to make sense of relativity using space and
time separately leads to effects such as time dilation
and length contraction
In the mathematical treatment of relativity, spacetime and energy-momentum objects are always
considered together
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Summary of Special Relativity
• The laws of physics are the same for all inertial
observers (inertial reference frames).
• The speed of light in vacuum is a universal
constant, independent of the motion of source
and observer.
• The space and time intervals between two events
are different for different observers, but the
spacetime interval is invariant.
• The equations of Newtonian mechanics are only
“non-relativistic” approximations, valid for
speeds small compared to speed of light.
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Everything Follows
• Lorentz transformation equations
• Doppler shift for light
• Addition of velocities
• Length contraction
• Time dilation (twin paradox)
• Equivalence of mass and energy (E=mc2)
• Correct equations for kinetic energy
• Nothing can move faster than c
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http://pujayanto.staff.fkip.uns.ac.id/
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May God Bless us
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