INST 240

RevolutionsLecture 10

Momentum and Energy

Which of the following is not associated with a direction, i.e. is

a number, not a vector?

• A: velocity

• B: momentum

• C: Force

• D: Mass

• E: acceleration

Which of the following IS associated with a direction, i.e. is

a vector?

• 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

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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)

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Kinetic Energy

Kinetic Energy =

The energy of a moving object.This is the form of energy discussed in spacetime diagrams in the book.

mass velocity squared

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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

Conservation of 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.

High gravitational, low kinetic energy

Low gravitational, high kinetic energy

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Conservation of Energy

Total initial energy = Total final energy

Putting a bucket of water on top of a doorInitial energy:

Gravitational potential energy

Final energy:

Kinetic energy

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Conservation of Energy

Total initial energy = Total final energy

Setting off a Bomb

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Conservation of Energy

Total initial energy = Total final energy

Setting off a Bomb

Chemical potential energy

= Heat, kinetic energy of debris, sound, light

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 GramObject/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

Uranium 235 20 million 30 million

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

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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.

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 = m V = m c

• 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

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The spacetime momentum vector

• Total length: mc• Length in time direction: γmc• Length in space direction: γmv = γ p

•

•

Pspace

Ptime

mc

γmv

γmc

Relativistic Formulae

• How do we change the non-relativistic formulae into the correct relativistic ones?

• Need to recover “old” formulae in limit that v0, i.e. for small velocities

• Note: γ 1 as v0

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 non-relativistic 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

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The correct formulae

• Due to length contraction and time dilation, momentum and energy equations change a little bit.

“Classical” Physics:

E = ½mv2

p = mv

Relativistic Physics

E = γmc2

p = γmv

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Non-relativistic limit• At low velocities (v << c):

“Classical” Physics:

E = ½mv2

p = mv

Relativistic Physics

E = γmc2

p = γmv→ mv + small corrections

→ mc2 + ½m v2+ small corrections