http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 1

Special relativity

•  Homework scores are on CULearn

•  Homework set 3 is on the website and is due Wed at 1:0pm.

Announcements:

Today we will continue with relativistic energy and the relationship between mass, momentum, and energy.

http://www.colorado.edu/physics/phys2170/

Cockcroft-Walton Accelerator

Physics 2170 – Fall 2013 2

Breakdown voltage in dry air is 30kV/inch. Above you see a 750kV difference – a little more than 25 inches for safety!

http://www.colorado.edu/physics/phys2170/

Homework #1 Distribution

Physics 2170 – Fall 2013 3

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 4

At low speeds, this becomes the familiar

Total energy:

Relativistic momentum and energy

At rest, γu=1 so the rest energy is

Kinetic energy is always the total energy minus the rest energy:

Relativistic momentum:

With these definitions for momentum and energy, conservation of momentum and conservation of (total) energy continue to work in isolated relativistic systems

This tells us that mass and energy are equivalent. Mass is a type of energy like motion is a type of energy (kinetic energy).

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 5

Rest Energy is real: Nuclear fission

Initial mass of a neutron and a 235U nucleus.

Final products have less mass, but much more kinetic energy. Conversion of mass to kinetic energy.

Oh yes, and more neutrons, so the reaction can run wild (chain reaction!).

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 6

Q. If we could convert mass entirely to energy, how much mass would be required to run a 30W light bulb for a year (a year is approximately 3•107 s)?

A. 10 kg B. 3 g C. 3 mg D. 10 µg E. 10 ng

Set frequency to AD Clicker question 1

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 7

Q. If we could convert mass entirely to energy, how much mass would be required to run a 30W light bulb for a year (a year is approximately 3•107 s)?

One year is 3•107 s so running a 30 W light bulb for a year requires E = P•Δt = 30 W • 3•107 s = 9•108 J. To get that energy from rest mass we use E=mc2 and solve for mass.

A. 10 kg B. 3 g C. 3 mg D. 10 µg E. 10 ng

Set frequency to AD Clicker question 1

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 8

A relativistic collision problem Object A has mass 9m0 and speed vA=0.8c (γA=5/3). Object B has mass 12m0 and speed vB=−0.6c (γB=5/4). The objects collide and stick together (completely inelastic collision)

Classically, what is the total initial momentum?

From Physics 1110 we know collisions conserve momentum and that inelastic collisions do not conserve kinetic energy

But this is an isolated system, so total energy must be conserved.

Let’s start with momentum conservation.

What is the total relativistic momentum?

So it does not end at rest as predicted classically!

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 9

A relativistic collision problem Object A has mass 9m0 and speed vA=0.8c (γA=5/3). Object B has mass 12m0 and speed vB=−0.6c (γB=5/4). Momentum conservation gives us:

Remember that mf may not be mA+mB as it would be classically.

Now let’s look at the total energy. The initial energy is

So conservation of energy gives us:

Dividing these two equations: or so

Furthermore:

conservation of energy equation for mf:

so we can solve the

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 10

Classically, total momentum is 0 but in reality it is

A relativistic collision problem Object A has mass 9m0 and speed vA=0.8c (γA=5/3). Object B has mass 12m0 and speed vB=−0.6c (γB=5/4).

Classically, but in reality, so 8.85m0 of mass is gained.

Classically, final kinetic energy is 0 and the initial kinetic energy is

So the change in KE is

But in reality, the initial and final kinetic energies are:

The “lost” kinetic energy appears as gained mass in the total energy

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 11

Really, mass gets created! CERN in Geneva, Switzerland

Before the LHC (Large Hadron Collider) CERN operated LEP, the Large Electron-Positron collider in the same underground tunnel.

Electron and positrons have a mass of 9x10-31 kg. They were accelerated to very high energies so when they annihilate, they create a Z0 particle with a mass of 1.6x10-25 kg.

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 12

Total energy: Relativistic momentum and energy

Relativistic momentum:

With some algebra we can eliminate the velocity variable from these two relations

A proton has a mass of 938 MeV/c2. What is this in kg?

A new unit of energy is the electron-volt (eV). It’s the energy obtained by an electron moving through 1 V. It is not an SI unit but is very common. ΔE = qΔV = 1 eV = 1.6•10-19 C • 1 V = 1.6•10-19 J

Also use eV/c or MeV/c units for momentum

Since mc2 is a unit of energy, dividing energy by c2 gives a unit of mass. Also, dividing energy by c gives a unit of momentum.

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 13

Set frequency to AD Clicker Question 2

True or false: A massless particle has a finite momentum.

A.  True B.  False C.  More than one of the above D.  None of the above

Please answer this question on your own.

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 14

Conservation of energy requires that the sum of the photon energies equal the total π0 energy.

Although we don’t need it, we can see that the π0 kinetic energy is KE = E – mc2 = 700 MeV – 135 MeV = 565 MeV

Can get velocity from From

we note that

A π0 particle decays into two photons. The photon energies are measured to be 200 MeV and 500 MeV. The π0 rest mass is 135 MeV/c2. What is the π0 velocity?

Conservation of energy for subatomic particles

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 15

What is the velocity of a π0 with rest mass 135 MeV/c2 and total energy 700 MeV?

We found before that we could solve

So this gives us:

So the π0 velocity was 0.98c

Conservation of energy for subatomic particles

to get