Transcript Monday, January 31 - Otterbein University

INST 240
Revolutions
Lecture 10
Momentum and Energy
Which of the following is not
associated with a direction, i.e. is
a number, not a vector?
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•
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A: velocity
B: momentum
C: Force
D: Mass
E: acceleration
Which of the following IS
associated with a direction, i.e. is
a vector?
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A: Energy
B: Position
C: Mass
D: Temperature
E: Time
Invariants and constants
• Not the same thing!
• Invariants are quantities different observers
agree on
• Constants are quantities that stay the same
for one observer, but another observer may
not agree on the value - or that it stays the
same
Examples
• The position of an object at rest is constant for an
observer at rest wrt the object, but not for a moving
observer
• The spacetime distance between two events is
constant and invariant (if the two events are fixed)
• The total momentum of a system (its mass time its
velocity) is a constant (unless a force acts on it) but it
is not an invariant
We need spacetime not space
vectors!
• Velocity is a space vector
• Momentum is a space vector
• Total momentum is conserved, so the total
momentum vector is conserved
• Not good enough for relativity: observers
will not agree on the length or direction of a
space vector, only of a spacetime vector!
What is Energy
• Work- energy theorem
• Energy is the ability to do work
Energy
•Roughly, the ability of a thing to influence other things
(technically, to “do work” on things)
– Example: drop a brick on your toe
•Energy is a number
•Comes in many forms (not all different!):
– Motion (“kinetic”)
– Gravitational
– Elastic
– Thermal (aka “heat”)
– Chemical
– Nuclear
– Electrical
– Radiant (light)
8
Kinetic Energy
The energy of a moving object.
This is the form of energy discussed in spacetime
diagrams in the book.
Kinetic Energy =
mass velocity squared
9
Other forms of energy
• Rotational kinetic energy - something is moving
• Thermal energy - atoms moving around when
something is hot
• Electromagnetic energy - light, radio, etc
• Electrical energy or Magnetic energy
• Chemical energy - fuel and air, energy bound
between atoms
• Nuclear energy - energy bound inside atoms
10
Conservation of Energy
High gravitational,
low kinetic energy
Energy can be converted from
one type to another, but cannot
be created or destroyed. The
total amount of energy in the
universe never changes.
Low gravitational,
high kinetic energy
Conservation of Energy
Total initial energy = Total final energy
Putting a bucket of water on
top of a door
Initial energy:
Gravitational potential energy
Final energy:
12
Kinetic energy
Conservation of Energy
Setting off a Bomb
Total initial energy = Total final energy
13
Conservation of Energy
Setting off a Bomb
Total initial energy = Total final energy
Heat,
Chemical
kinetic energy of
potential
debris,
energy
sound, light
=
14
Measuring Energy
1 Joule (official scientific unit; apple lifted 1 meter)
1 Calorie (food) = 4200 Joules (heat 1 kg water by 1ºC)
1 Jelly Donut (JD) = 250 Calories or 106 Joules ($0.75)
Typical American diet = 10 JD per day
1 kilowatt-hour (kWh) = 3.6 million Joules
($0.09)
1 gallon of gasoline provides 30,000 Calories ($3.00)
1 Megaton TNT (large nuclear weapon) = 1012 Calories
Energy per Gram
Object/Material
Calories
Compared to TNT
Bullet (moving at sound speed, 1000 ft/sec)
0.01
0.015
Battery (car)
0.03
0.05
Battery (rechargeable computer)
0.1
0.15
Battery (alkaline flashlight)
0.15
0.23
TNT
0.65
1
Modern high explosive (PETN)
1
1.6
Chocolate chip cookies
5
8
Coal
6
10
Butter
7
11
Alcohol (ethanol)
6
10
Gasoline
10
15
Natural gas (methane, CH4)
13
20
Hydrogen (H2)
26
40
Asteroid or meteor (moving at 30 km/sec)
100
165
20 million
30 million
Uranium 235
Tunguska
• ~ 30 m diameter body struck
Siberia on June 30, 1908
• Detonation above ground; no
obvious crater(s)
• Destroyed about 800 square
miles of forest; heard 500 mi
away
• Houses destroyed 200 mi
away
• Dust appeared in London,
6,200 mi away
If Tunguska had been London
Why did it explode?
• An explosion happens when a large amount
of stored energy is converted to heat (really
another form of energy) in a small space
• Nearby stuff vaporizes, turning into hot gas
with high pressure
• The hot gas expands rapidly, pushing other
stuff out of the way
• The flying debris is typically what causes the
damage in an explosion
Energy can be transformed into
other types
• Potential energy (object at greater height) to
kinetic energy (moving object): Video
• Chemical energy into potential energy,
kinetic energy, deformation, heat, sound,
radiation, etc.: Video
• Nuclear binding energy into heat, potential
& kinetic energy, radiation, etc. Video
Spacetime momentum
• We are using spacetime graphs to represent
where events happened and when they
happened.
• We want to use spacetime graphs to represent
the momentum of objects, too.
21
Space position & c times time =
spacetime position
• Space momentum & ? = spacetime
momentum
• Need to find something “similar”, related to
momentum by speed of light
How do we define velocity
relativistically correct?
• Problem: we cannot agree on the distance
traveled nor the time elapsed!
• In spacetime things are easier
– We agree on the spacetime distance Δs
– We agree on the elapsed proper time Δ t = Δ s/c
• But: velocity in spacetime is
Δs/Δt = Δs/Δs*c = c
Remember the motorcyclist!
Relativistically correct time aka
Proper time
Spacetime velocity
• Standard definition of velocity: v = distance
per elapsed time = Δx/Δt
• Einstein: No good! Observers do not agree
on distance or time
• Replace Δx with spacetime distance Δs
• Replace time with proper time: Δt Δs/c
• Then V = Δs/(Δs/c) = c
• The spacetime velocity is a constant c!
Not as boring as it seems!
• The LENGTH of the spacetime velocity is c
• The DIRECTION of the spacetime velocity
depends on the motion itself
– Points from initial position and time to final
position and time
– Example: baseball rolled at noon to the right
Upgrade velocity to momentum
• Simply multiply by mass:
P=mV=mc
• Wait, what about direction?
• This is “built in”, the direction in spacetime is
pointing from the initial event to the final event
• Baseball being at origin at noon, 2m to the
right 2 seconds later has p = 0.1 kg m/s but P =
0.1 kg c = 30,000,000 kg m/s
Spacetime momentum vector
• Always points in the direction the object
travels
• Has length or magnitude: mc, since
spacetime velocity is a constant c
Split into space and time direction
• How much of the spacetime momentum P
point in space direction?
• Take its space part Δx, divide by proper
time Δs/c = Δt/γ and multiply by mass:
Pspace = m γ Δx/Δt = γ m v
• Analogously for time:
Ptime = m γ cΔt/Δt = γ m c
The spacetime momentum vector
Ptime
mc
γmv
γmc
Pspace
• Total length: mc
• Length in time direction: γmc
• Length in space direction: γmv = γ p
30
Relativistic Formulae
• How do we change the non-relativistic
formulae into the correct relativistic ones?
• Need to recover “old” formulae in limit that
v0, i.e. for small velocities
• Note: γ  1 as v0
Taking Momentum Conservation
seriously
• If we do, then space and time part of
spacetime momentum should be conserved
independently!
• For space part no problem: in nonrelativistic world γ  1, so
Pspace = γmv  mv = p
Taking Momentum Conservation
seriously
• Time part: γmc is conserved, so γmc2
(which has units of energy) is conserved!
• Well, what is that?
• In non-relativistic limit γ  1
• More precisely
γ = 1 + (v/c)2 + 3/8 (v/c)4 + …
≈ 1 + (v/c)2
The correct formulae
“Classical” Physics:
E = ½mv2
p = mv
Relativistic
Physics
E = γmc2
p = γmv
• Due to length contraction and time dilation,
momentum and energy equations change a
34 little bit.
Non-relativistic limit
• At low velocities (v << c):
“Classical”
Physics:
E = ½mv2
p = mv
35
Relativistic Physics
E = γmc2 → mc2 + ½m v2+
small corrections
p = γmv → mv + small
corrections