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

Special relativity

•  Homework solutions will soon be CULearn

•  Homework set 1 returned today. •  Homework #2 is due today. •  Homework #3 is posted – due

next Wed. •  First midterm is 2 weeks from

tomorrow.

Announcements:

Today we will investigate the relativistic Doppler effect and look at momentum and energy.

Christian Doppler (1803—1853)

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

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Physics 2170 – Fall 2013 2

Need to register your clickers for me to be able to associate scores with you

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 3

Set frequency to AD Clicker question 4 last lecture A spacecraft travels at speed v=0.5c relative to the Earth. It launches a missile in the forward direction at a speed of 0.5c. How fast is the missile moving relative to Earth?

A.  0 B.  0.25c C.  0.5c D.  0.8c E.  c

Have to keep signs straight. Depends on which way you are transforming. Also, the velocities can be positive or negative!

Best way to solve these is to figure out if the speeds add or subtract and then use the appropriate formula.

This actually uses the inverse transformation:

Since the missile if fired forward in the spacecraft frame, the spacecraft and missile velocities add in the Earth frame.

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

Velocity addition works with light too! A Spacecraft moving at 0.5c relative to Earth sends out a beam of light in the forward direction. What is the light velocity in the Earth frame?

What about if it sends the light out in the backward direction?

It works. We get the same speed of light no matter what!

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Relativistic Doppler Effect •  Train (frame S’)

traveling at speed v relative to the ground (frame S). Headlight from train is a source of light with frequency fsource. Observer is on the ground at rest in front of the train + we want to determine the observed frequency, fobs.

Physics 2170 – Fall 2013 5

The time between emission of two crests is given by Δt. During this time first crest will move a distance cΔt, but the train will move a distance vΔt.

Distance between successive crests is therefore λ = cΔt – vΔt (As seen from S)

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Doppler cont.

•  Since all crests are approaching at speed c, the frequency seen by observer Q, fobs , is

Physics 2170 – Fall 2013 6

fobs = c/λ = c/(c-v)Δt = 1/(1-β)Δt

Δt is time between successive crests as measured in S, but Δt = γ Δt’ with Δt’ being the proper time between two crests, since they occur at same place in S’.

Now fsrc = 1/Δt’, hence

fobs = fsrc/[ γ(1-β)]

1/γ =√(1-β2 = √(1-β)(1+β)

€

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

Relativistic Doppler shift The speed of light is the same for all inertial observers

However, the wavelength and frequency change based on relative velocity

For a source moving toward an observer:

For a source moving away switch + and -

It does not matter if it is the source or the observer that is moving; only the relative velocity matters.

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

Set frequency to AD Clicker question 1 An alien on his spaceship sends a laser beam toward Earth using a special green laser pointer. The people on Earth observe a yellow light from the alien spaceship. Is the spaceship moving toward or away from Earth? A. Spaceship is headed to Earth B. Spaceship is headed away from Earth C. Impossible to tell

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

Set frequency to AD Clicker question 1 An alien on his spaceship sends a laser beam toward Earth using a special green laser pointer. The people on Earth observe a yellow light from the alien spaceship. Is the spaceship moving toward or away from Earth? A. Spaceship is headed to Earth B. Spaceship is headed away from Earth C. Impossible to tell

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

Relativistic Doppler shift Since c is constant and c=λf then

For approaching source, λ is shorter – blueshift

For receding source, λ is longer – redshift In 1929 Hubble showed the velocity of

galaxies (measured using redshift) was proportional to distance. First evidence for the Big Bang theory.

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

Relativistic Doppler shift

Used to measure velocity in police and baseball radar guns.

Used in Doppler radar to measure the speed of the air/rain.

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

Moving from kinematics to dynamics In Physics 1110 we began by discussing velocities and accelerations and called this kinematics.

Then we moved to Newton’s laws of motion which tells us that it is force that causes acceleration. This is called dynamics.

Finally, we used conservation of momentum and conservation of energy to avoid the complication of calculating accelerations (as long as we had an isolated system).

Let’s start thinking about momentum:

Classically, momentum is p=mu where we continue using u to represent the velocity of an object while v represents the velocity of a frame.

What we really need momentum for is to use conservation of momentum on problems like collisions and explosions.

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

Conservation of momentum Conservation of momentum states that for an isolated system (no net force):

What if we observe this isolated system in a different inertial reference frame? Using Galilean transformations we get (in 1D)

so that

This just says that the momentum changes by the mass of the system times the relative velocity v.

The velocity between these two inertial reference frames (v) is constant and mass is constant so if momentum is conserved in one inertial reference frame (Ptotal) then it is conserved in all inertial reference frames (P′total).

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

We defined momentum as

Conservation of momentum But we know that the Galilean transformations are not correct at high velocity. If we apply the correct transformations we find that if momentum is conserved in one reference frame it is not necessarily conserved in other inertial reference frames.

So we need a new definition of momentum.

We try

We know that Δt depends on which inertial frame you are in but there is one time that stays the same: the proper time. This is the time measured in the rest frame and we will know call it tau (Δτ).

and remember time dilation:

This gives us:

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

Conservation of momentum So the relativistic momentum is:

Note the addition of a subscript on γ.

Our previous use of γ was to relate between two different frames with a relative velocity of v. In contrast, γu is associated with a particle. If we measure p=γumu in one inertial frame we can convert the momentum to another inertial reference frame moving with speed v which will introduce another γ which we should probably call γv.

It should be clear by context which one we are talking about so I will probably drop the subscript after a while.

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 16

A B

Particle A has half the mass but twice the speed of particle B. If the particles’ momenta are pA and pB, then

A. pA > pB

B. pA = pB

C. pA < pB

Set frequency to AD Clicker question 2

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 17

A B

Particle A has half the mass but twice the speed of particle B. If the particles’ momenta are pA and pB, then

A. pA > pB

B. pA = pB

C. pA < pB γu is bigger for the faster particle.

Set frequency to AD Clicker question 2

Classically, both particles have the same momentum.

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 18

So let us postulate that energy is

Momentum transformation and energy

Using the relativistic momentum and the correct velocity we find:

What other quantity is conserved when no external forces act?

Using the old momentum and Galilean transformation to get from S to S′ frame:

Since v and γv are constants, in order to have conservation of momentum in each frame, the quantity γum must also be constant.

Energy! γum has units of mass (kg); to give it units of energy, can multiply by c2 (which we know is constant).

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 19

Total energy of an object moving at speed u is

Energy

What do we get for the total energy when an object is at rest? At rest, γu=1 so the rest energy is Maybe you have

heard of this one before?☺ Furthermore, we can define kinetic energy

as the total energy minus the rest energy:

Remember the binomial approximation for γ is

Using this on the kinetic energy gives:

(for small β)

So we get the correct kinetic energy at low speed.

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 20

u c

u c

u c

u c

E

E

E

E

Which of the graphs below is a possible representation of the total energy of a particle versus its speed

A. B.

C. D.

E. None of them

Set frequency to AD Clicker question 3

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 21

u c

u c

u c

u c

E

E

E

E

Which of the graphs below is a possible representation of the total energy of a particle versus its speed

A. B.

C. D.

E. None of them

Set frequency to AD Clicker question 3

As the velocity u increases towards c, the energy increases. But for u→c, γ→∞ so it is impossible for a particle to reach c as it would need infinite energy.

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 22

At what speed is the total energy of a particle equal to twice its rest mass energy? A. 0

B. 0.7c

C. 0.87c D. 0.94c E. c

Set frequency to AD Clicker question 4

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 23

At what speed is the total energy of a particle equal to twice its rest mass energy? A. 0

B. 0.7c

C. 0.87c D. 0.94c E. c

To have total energy equal to twice the rest mass energy, need γ=2

Set frequency to AD Clicker question 4

Solve for β.

so you need to be moving pretty fast to get your kinetic energy close to your rest mass energy!