# lect28 handout

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• 1. Special Relativity Physics 102:Lecture 28

2. Inertial Reference Frame

• Frame which is in uniform motion (constant velocity)
• No Accelerating
• No Rotating
• Technically Earth is not inertial, but its close enough.

3. Postulates of Relativity

• Laws of physics are the same in every inertial frame
• the same laws of physics can be used in any inertial reference frame and the results will be consistent for that frame.
• Speed of light in vacuum iscfor everyone
• Measure c=3x10 8m/s if you are on train going east or on train going west,even if light source isnt on the train.

Weird! 4. Relative Velocity (Ball)

• Josh Beckett throws baseball @90 mph. How fast do I think it goes when I am:
• Standing still?
• Running 15 mph towards?
• Running 15 mph away?

90 mph 90+15=105 mph 90-15=75 mph Example (Review 101 for help with Relative Velocities) 5. Relative Velocity (Light)

• Now he throws a photon (c=3x10 8m/s). How fast do I think it goes when I am:
• Standing still
• Running 1.5x10 8m/s towards
• Running 1.5x10 8m/s away

Strange but True! 3x10 8m/s 3x10 8m/s 3x10 8m/s Example Preflight 28.1 6. Consequences:1.Time Dilation t 0is call the proper time.Here it is the time between two events that occur at the same place, in the rest frame. D 7. Time Dilation t 0is proper time Because it is rest frame of event D D L=v t v t 8. Importantly, when deriving time dilation the two events, 1) photon leaving right side of train and2) photon arriving back at right side of train, occurat same point on the train . 9. Time Dilation A (pion) is an unstable elementary particle.It may decay into other particles in 10 nanoseconds. Suppose a +is created at Fermilab with a velocity v=0.99c.How long will it live before it decays? Example

• If you are moving with the pion, it lives 10 ns
• In lab frame where it has v=0.99c, it lives 7.1 times longer
• Both are right!
• This is not just theory.It has been verified experimentally
• (many times!)

10. Time Dilation Example v/c 0.1 1.005 0.2 1.021 0.5 1.155 0.9 2.294 0.99 7.089 0.999 22.366 0.9999 70.712 0.99999 223.607 0.999999 707.107 0.9999999 2236.068 11. Derive length contraction using the postulates of special relativity and time dilation. Consequences:2.Length contraction 12. (ii) Train traveling to right with speed v t 0= 2 x 0/c send photon to end of train and back the observeron the groundsees : forward backward where t 0= time for light to travel to front and back of train for theobserver on the train. x 0, = length of train according to theobserver on the train. note:see D. GriffithsIntroduction to Electrodynamics (i) Aboard train t forward=( x + v t forward)/c= x/(c-v) t backward=( x v t backward)/c= x/(c+v) t total= t forward+ t backward = (2 x/c ) 2= t 0 where we have used time dilation: t tota l = t 0 2 x = c t 0 = 2 x x = x / the length of the moving train is contracted! 13. Space Travel Alpha Centauri is 4.3 light-years from earth. (It takes light 4.3 years to travel from earth to Alpha Centauri).How long would people on earth think it takes for a spaceship traveling v=0.95c to reach Alpha Centauri? How long do people on the ship think it takes? People on ship have proper time. They see earth leave, and Alpha Centauri arrive in t 0 t 0= 1.4 years Example 14. Length Contraction People on ship and on earth agree on relative velocity v = 0.95 c.But they disagree on the time (4.5 vs 1.4 years).What about the distance between the planets? Earth/AlphaL 0= v t = .95 (3x10 8m/s) (4.5 years) = 4x10 16 m(4.3 light years) ShipL = v t 0 = .95 (3x10 8m/s) (1.4 years) = 1.25x10 16 m(1.3 light years) Example Length in moving frame Length in objects rest frame 15. Length Contraction Notice that even though the proper time clock is on the space ship, the length they are measuring is not the proper length. They see a moving stick of length L with Earth at one end and Alpha-Centauri at the other. To calculate the proper length they multiply their measured length by . Length in objects rest frame Example Length in moving frame 16. Length Contraction Sue is carrying a pole 10 meters long.Paul is on a barn which is 8 meters long.If Sue runs quickly v=0.8 c, can she ever have the entire pole in the barn? Paul: The barn is 8 meters long, and the pole is only Sue: No way! This pole is 10 meters long and that barn is only Example Who is right? A) PaulB) SueC) Both 17. ACT / Preflight 28.3 Youre at the interstellar caf having lunch in outer space - your spaceship is parked outside.A speeder zooms by in an identical ship at half the speed of light.From your perspective, their ship looks: (1) longer than your ship (2) shorter than your ship (3) exactly the same as your ship 18. Time Dilation vs. Length Contraction

• Time intervals between same two events :
• Consider only those intervals which occurat one point in rest frameon train.
• t ois in the reference frame at rest, on train.proper time
• t is measured between same two events in reference frame in which train is moving, using clock that isnt moving, on ground, in that frame.
• Length intervals of same object:
• L 0 is in reference frame where object is at restproper length
• Lis length of moving object measured using ruler that is not moving.

L o> L Length seems shorterfrom outside t > t o Time seems longerfrom outside 19. Consequence: Simultaneity depends on reference frame (i) Aboard train Light is turned on and arrives atthe front and back of the train at the same time.The two events,1) light arriving at the front ofthe train and2) light arriving at theback of the train,aresimultaneous . (ii) Train traveling to right with speed v Light arrives first at back of the train and then at the front of the train, because light travels at the same speed in all reference frames. The same two events,1) light arriving at the front ofthe train and2) light arriving at theback of the train,areNOT simultaneous . the observeron the groundsees : note:see D. GriffithsIntroduction to Electrodynamics 20. Therefore whether two events occur at the same time-simultaneity- depends on reference frame. 21. Relativistic Momentum Relativistic Momentum Note: for v

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