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INST 240 Revolutions Lecture 9 Directions in Spacetime, Twin Paradox & Vectors

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Spacetime Motorcycle Analogy Crazy train: The space time motorcycles can't go East or West because he has to stay at the fixed line set at 45 degrees relative to the speed of light. The motorcycles throttle is fixed because the faster the 'bike' goes the observers will think the bike is going at a different pace. Space and time must always stay the same. Somebody: The motorcyclist cannot travel East or West in spacetime because he would have to be traveling faster than the speed of light, or the cosmic speed. Also, he would be changing his position in space, without any time elapsing. The throttle is fixed because everything in spacetime has to move at the same speed, including the motorcyclist.

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Spacetime Motorcycle Analogy Motorcycle rolling through undulating landscape with criss-crossing roads Assume traveling due north is like traveling in time direction All roads lie within a bearing of 45 degrees of north

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Everything travels through spacetime at the same speed c Everything? Yes! But spacetime is not space! So the motorcycle’s throttle is fixed Think about ct: even as you sit still, time passes and the s grows 1 light-second every second

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Proper time As you sit still, your clock shows a special time: the time for an observer at rest It directly translates into spacetime distance, since no space distance x is involved The wristwatch measures spacetime distance: s= ct ! This time is called “proper time” and everyone agrees on its value – since everyone agrees on c and s!

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Moving clocks waste their quota of spacetime speed by moving! This is why they run slower: the move around in space and therefore become slower in the time direction

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Worksheet Please convene in groups to do the ranking problem on the worksheet.

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8 Twin Paradox Two twins, Cindy and Sally. –Cindy stays home –Sally becomes an astronaut and travels on a rocket for a few years, takes a few pictures, and then comes home.

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9 What’s the paradox? - WarmUp Crazy Train The paradox is one twin is moving at the speed of light slowing down her life while the other twin is sitting back on Earth having the same normal life time. The contraindication is that there is no absolute motion; saying either one of the twins could be moving its all in the matter of who the observer choses. Somebody:The two seemingly paradoxical statements are that each twin could say that the other one would be older when the astronaut twin returns to earth. In other words, based on the frame of reference, each twin would view herself to be younger. In reality, though, there is no paradox.

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10 Sally leaves Sally Ride, first woman in space Both start at age 25. Cindy stays home. Cindy sees Sally move away at 3/4 c. Sally goes to space Sally sees Cindy move away at 3/4 c. So far, no paradox. EarthAndromeda

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11 Sally leaves Both start at age 25. Cindy stays home. Cindy sees Sally’s clock slow down by a factor of 1.5. Sally goes to space Sally sees Cindy’s clock slow down by a factor of 1.5. So far, no paradox. EarthAndromeda

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12 Sally leaves Sally Ride, first woman in space Both start at age 25. So far, no paradox - they are in different places, so they mean different things by this. 4 years pass for Cindy. 2.7 years pass for Sally. 2.7 years pass for Cindy. 4 years pass for Sally.

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13 Sally Comes Home. Cindy sees Sally’s clock slow down by a factor of 1.5. Cindy got old, Sally stayed young. EarthAndromeda Total 8 years pass for Cindy. Total 5.3 years pass for Sally.

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14 Sally Comes Home Both start at age 25. Sally got old, Cindy stayed young. Earth Andromeda Sally sees Cindy’s clock slow down by a factor of 1.5. Total 5.3 years pass for Cindy. Total 8 years pass for Sally.

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15 Paradox When Sally gets home, the two twins can look at each other, and see: who was right? They can’t both be right - someone has to be older!

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16 The Resolution - Definition of an Inertial Frame 16 Somebody: An "inertial reference frame" implies that time and space are the same and that there is no acceleration. However, the traveling twin is accelerating, so she would not be in an "inertial reference frame."

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17 The Resolution - Definition of an Inertial Frame 17 Inertial reference frame is: A frame of reference moving at constant velocity. “Physics is invariant if you are in an inertial reference frame.. i.e. moving at constant speed.”

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Worksheet Please convene in groups to do the spacetime distance problem on the worksheet.

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19 There are actually 3 inertial frames 19 Rocket earth Rocket earth Earth frame Frame of rocket on outbound flight. Frame of rocket on return flight.

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20 There are actually 3 inertial frames 20 Roc ket earth Roc ket earth Earth frame earth There is no frame which holds the rocket through it’s whole flight - the laws of physics for the rocket are NOT invariant.

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21 O×O× Rocket earth Earth frame s = 8 ly ct = 8 ly x = 0 s = 2.7 ly ct = 4 ly x = 3 ly s = 2.7 ly ct = 4 ly x = 3 ly Cindy (earth) waits 8 years for her twin to return. Meanwhile, Sally travels at 3/4 the speed of light for 4 years, then another 4 years on the return trip.

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22 O×O× Rocket earth Earth frame s cindy = 8 ly s sally = 5.2 ly Answer: Remember, we can interpret s as the proper time. So, it took Cindy 8 years for the trip. It took Sally 5 years for the trip. Sally comes back looking young!

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23 Rocket earth Roc ket earth If we worked out the answer in the other two frames, we would get the same answer: Sally ages a little, Cindy ages a lot.

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Arrows & Vectors The arrows in the spacetime diagrams are a version of what scientists call a vector Vectors are more powerful than numbers –They have a length (number, magnitude) –They point in a direction Vectors in spacetime and vectors in space are different, but the same type of thing

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Kinematics & Dynamics So far we have been describing HOW things are moving (hard enough to agree on a good & valid description) This is called Kinematics Now we want to understand WHY things are moving This is called Dynamics

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Kinematics Words Position Velocity (Rate change of position ) Acceleration (Rate change of velocity) Time

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Dynamics Words Mass Momentum (mass times velocity) Force (rate change of momentum – or imagine a push or pull in a certain direction)

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Demo: (Force) Vector Old-fashioned force-meters

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29 Mass Mass is a measure of two things: –How difficult a thing is to lift against gravity –How difficult a thing is to get moving. –The latter is more important to us today. Scientists used to think (until Einstein) that mass is never created or destroyed. Total initial mass = Total final mass

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30 Momentum Formally, momentum is: p = mv Momentum = mass times velocity Moving things have momentum. Faster things have more momentum. Heavier things have more momentum. Momentum is both a quantity and a direction, physicists call objects like this a vector.

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Demo: Momentum Conservation Two billiard balls Two carts on the track

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32 Momentum is Conserved 2 lbs 1 m/s Total momentum = 0 Total initial momentum = total final momentum

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33 Momentum is Conserved 2 lbs 1 m/s Total momentum = 0 2 lbs 1 m/s Total momentum = 0 Total initial momentum = total final momentum

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34 Momentum is Conserved 2 lbs 1 m/s Total momentum = 0 2 lbs 0 m/s Total momentum = 0 Total initial momentum = total final momentum

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35 Momentum is Conserved Total initial momentum = total final momentum 2 lbs 1 m/s Total momentum = 0 2 lbs 1 m/s Total momentum = 0

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