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Special Relativity SPH4U

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Review of Scientific Theories Recall discussion from the first day of class Recall discussion from the first day of class A scientific theory is a proposed explanation/description for observed facts A scientific theory is a proposed explanation/description for observed facts It is possible for a theory to be a good approximation or have some usefulness even if it is not fully correct It is possible for a theory to be a good approximation or have some usefulness even if it is not fully correct One of the best examples is Newtonian Physics vs. Relativity & Quantum Mechanics One of the best examples is Newtonian Physics vs. Relativity & Quantum Mechanics

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Newtonian Physics Physics principles as explained by Newton and others Physics principles as explained by Newton and others Newtons 3 Laws and Law of Gravity Maxwells Equations of Electromagnetism Equations for motion, momentum, kinetic energy, etc. discussed earlier in this class Underlying foundations of space and time as absolute Underlying foundations of space and time as absolute

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Relativity and Quantum Mechanics New physics as described by Einstein and others, most of the work done in the early 1900s New physics as described by Einstein and others, most of the work done in the early 1900s Time dilation, length contraction Time dilation, length contraction Uncertainty principle Uncertainty principle Bohr Theory of the Atom Bohr Theory of the Atom Different fundamental assumptions about the Universe Different fundamental assumptions about the Universe

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The Special Theory of Relativity Aimed to answer some burning questions: Aimed to answer some burning questions: Could Maxwells equations for electricity and magnetism be reconciled with the laws of mechanics? Could Maxwells equations for electricity and magnetism be reconciled with the laws of mechanics? Where was the aether? Where was the aether?

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The Conflict Newtonian physics seems to describe the world as we are used to it Newtonian physics seems to describe the world as we are used to it However, several experiments as well as some hypothetical arguments signaled some problems However, several experiments as well as some hypothetical arguments signaled some problems Relativity and Quantum Mechanics improve upon Newtonian physics Relativity and Quantum Mechanics improve upon Newtonian physics

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Newtonian Physics Newtonian physics accurately describes the Universe when… Newtonian physics accurately describes the Universe when… Speeds are not too large Speeds are not too large Gravity is not too strong Gravity is not too strong You are at a macroscopic level, i.e. not dealing with individual molecules/atoms You are at a macroscopic level, i.e. not dealing with individual molecules/atoms

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Newtonian Physics, cont. Under the conditions of the previous slide, there is no reason to use anything other than Newtonian physics Under the conditions of the previous slide, there is no reason to use anything other than Newtonian physics Equations give the same results to high accuracy Equations give the same results to high accuracy Example: Trajectories of satellites and space probes use Newtonian physics Example: Trajectories of satellites and space probes use Newtonian physics

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Relativity Relativity is a set of physics concepts and laws deduced by primarily by Albert Einstein Relativity is a set of physics concepts and laws deduced by primarily by Albert Einstein Special Relativity Special Relativity Published by Einstein in 1905 Published by Einstein in 1905 Special case with no forces/acceleration Special case with no forces/acceleration General Relativity General Relativity Published by Einstein in 1915 Published by Einstein in 1915 Extension of previous theory to include forces Extension of previous theory to include forces

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What ISNT Relativity? Relativity does not simply mean everything is relative Relativity does not simply mean everything is relative On the contrary, relativity says certain things are relative, and other things are absolute On the contrary, relativity says certain things are relative, and other things are absolute Relativity also tells us by how much those certain things are relative and in what way Relativity also tells us by how much those certain things are relative and in what way

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Experimental and Theoretical Need for Relativity Michelson-Morley Experiment Michelson-Morley Experiment Speed of light is the same regardless of the Earths motion through the aether (absolute space) Speed of light is the same regardless of the Earths motion through the aether (absolute space) Maxwells Equations of Electromagnetism Maxwells Equations of Electromagnetism Predict very unusual things, like magnetic fields with loose ends, when speeds are extremely large Predict very unusual things, like magnetic fields with loose ends, when speeds are extremely large

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Michelson-Morley Experiment For a long time, scientists believed in an aether absolute space For a long time, scientists believed in an aether absolute space In the Michelson-Morley experiment, the speed of light was measured with and across the flow of the aether as the Earth moved through it In the Michelson-Morley experiment, the speed of light was measured with and across the flow of the aether as the Earth moved through it

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Michelson-Morley Experiment Flash Flash Flash Contrary to expectations, however, the speed of light was the same both with and across! Contrary to expectations, however, the speed of light was the same both with and across!

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Theoretical Foundations of Relativity To explain all of these things, Einstein came up with new laws of physics based on two assumptions To explain all of these things, Einstein came up with new laws of physics based on two assumptions The laws of physics are the same in all inertial (non- accelerating) frames The laws of physics are the same in all inertial (non- accelerating) frames The speed of light is the same as measured by all observers in all inertial frames The speed of light is the same as measured by all observers in all inertial frames Einstein took these principles on faith Einstein took these principles on faith The principles and their implications have passed subsequent experimental testing The principles and their implications have passed subsequent experimental testing

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Relatively Speaking What do Einsteins two assumptions imply? What do Einsteins two assumptions imply? All motion is relative All motion is relative Relativity of simultaneity Relativity of simultaneity Relativistic velocity addition Relativistic velocity addition Time dilation Time dilation Length contraction Length contraction Relativistic mass increase Relativistic mass increase E = mc 2 E = mc 2

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Who is moving?

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All Motion is Relative, cont. You and your friend Jackie like to travel in bizarre spherical spaceships You and your friend Jackie like to travel in bizarre spherical spaceships Who is moving? Who is moving? Who is stationary? Who is stationary?

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All Motion is Relative, cont. In spite of our everyday intuition, the only velocities that can be measured are relative velocities In spite of our everyday intuition, the only velocities that can be measured are relative velocities Examples: Examples: Relative to the surface of the Earth Relative to the surface of the Earth Relative to the Sun Relative to the Sun Relative to a distant galaxy Relative to a distant galaxy

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Galilean Relativity 1,000,000 ms -1 How fast is Spaceship A approaching Spaceship B? Both Spaceships see the other approaching at 2,000,000 ms -1. This is Galilean or Classical Relativity.

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Einsteins Special Relativity 1,000,000 ms -1 0 ms -1 300,000,000 ms -1 Both spacemen measure the speed of the approaching ray of light. How fast do they measure the speed of light to be?

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Nothing Can Go Faster Than The Speed of Light

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Addition of Velocities In normal circumstances, if you are moving and throw an object, an outside observer will see the object at a different velocity In normal circumstances, if you are moving and throw an object, an outside observer will see the object at a different velocity Straight-forward velocity addition Straight-forward velocity addition But all observers measure the speed of light to be the same But all observers measure the speed of light to be the same

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Velocity Additions Do Not Apply to Light Even if you are moving away from your friend at a very high velocity, you will both see a light beam moving at c. Even if you are moving away from your friend at a very high velocity, you will both see a light beam moving at c. Nice to know formula

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Relativistic Velocity Additions A formula for adding velocities exists, but it is not required for the course. A formula for adding velocities exists, but it is not required for the course. The formula works such that you can never get velocities greater than c The formula works such that you can never get velocities greater than c For small velocities, is approximately the same as just adding the velocities For small velocities, is approximately the same as just adding the velocities

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Relativistic Velocity Additions Relativistic Velocity Additions

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Relativity of Simultaneity Two lights an equal distance from M go off Two lights an equal distance from M go off A passing train carries M A passing train carries M M sees the light from B first M sees the light from B first M see the light flashes at the same time M see the light flashes at the same time M is moving in the direction of B M is moving in the direction of B This relativity is determined by the speed of light and the relative motion of the objects/observers This relativity is determined by the speed of light and the relative motion of the objects/observers

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Relativity of Simultaneity Events which are simultaneous in one frame may not be in another! Events which are simultaneous in one frame may not be in another! Each observer is correct in their own frame of reference Each observer is correct in their own frame of reference

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The Lorentz Factor Calculating length contraction, time dilation, and other quantities requires calculating the Lorentz factor Calculating length contraction, time dilation, and other quantities requires calculating the Lorentz factor = v/c = v/c If v = 99% of c, then = 0.99 If v = 99% of c, then = 0.99 is always < 1 1 is always < 1 1

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The Lorentz Factor, cont. Some examples: Some examples: v = 0.1% of c = 1.0000005 v = 0.1% of c = 1.0000005 v = 1% of c = 1.00005 v = 1% of c = 1.00005 v = 10% of c = 1.005 v = 10% of c = 1.005 v = 50% of c = 1.155 v = 50% of c = 1.155 v = 90% of c = 2.294 v = 90% of c = 2.294 v = 99% of c = 7.089 v = 99% of c = 7.089 v = 99.9% of c = 22.37 v = 99.9% of c = 22.37

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Time Dilation This time is known as Proper Time. Because the clock is rest the frame of the occurring event. The Proper Time interval between two events is always the time interval measured by an observer for whom the two events take place at the same position. Note : D A clock using light pulses to keep time. Every time the pulse returns, a unit of time has passed Distance = Velocity x Time

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V Time Dilation Now since D ½ v t ½ c t L=v t We are now watching the clock move horizontally with velocity v. We will examine one cycle, more specifically one-half of one cycle. During a cycle of the light photon the clock will have moved horizontally a distance L, and if we calculate the distance travelled by the light in this one cycle (a upside down V), the distance would be c times the time we measured for the cycle, that is ct. So for one-half cycle the distance travelled by the light is ½ ct. This says a moving clock run slow. If t s =1 then you watching it move would notice it taking more than 1s (t m >1) on your clock, so t s runs slow.

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Time Dilation Example You and a friend are having a eating contest. Your friend is on a train traveling at speed v=0.9 c. By her watch, she finishes her food in 5 seconds. Determine the time you measure, if you are standing still at the train station. Since eating is happening on the train, that is the proper time, t s =5.

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Time Dilation Example 2 Now it is your turn to eat. According to your watch you finish your food in 5 seconds. How long does your friend think it took you to finish the food? Now eating is happening at the station, so that is theproper time, again t s =5. Both people think they won! Your friend would consider you to be moving. Remember the proper time is where the event and clock are together

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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 A.C.? How long do people on the ship think it takes? People on ship have proper time since they see earth leave, and Alpha Centauri arrive. t s t s = 1.4 years

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Space Travel Another approach that solves any special relativity problem by treating space and time as spacetime. The only requirement is that both separated units are recorded in the same units. (i.e.: light seconds, light minutes, light years, …)

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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 A.C.? How long do people on the ship think it takes? An amazing technique is to place time and space in the same units then use the following relativistic formula:

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Time Dilation Review Time flows more slowly in a moving frame as observed by an outside observer Time flows more slowly in a moving frame as observed by an outside observer But remember motion is relative But remember motion is relative If you and I are moving past each other If you and I are moving past each other I see your clock moving more slowly But you also see mine moving more slowly…!!!

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Length Contraction Objects moving relative to an outside observer appear contracted in the direction of their motion as measured by the observer Objects moving relative to an outside observer appear contracted in the direction of their motion as measured by the observer

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Length Contraction, cont. If you and I move past each other in some sweet sports cars If you and I move past each other in some sweet sports cars I measure your sports car as being shorter You measure my sports car as being shorter Only applies to the direction of motion Only applies to the direction of motion We see our sports cars as still being the same height

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Length Contraction v=0.1 c v=0.8 c v=0.95 c

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Length Contraction Example 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/Alpha: d 0 = v t=.95 (3x10 8 m/s) (4.5 years) = 4x10 16 m (4.3 light years) Ship: d = v t =.95 (3x10 8 m/s) (1.4 years) = 1.25x10 16 m (1.3 light years) Length in moving frame Length in objects rest frame

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Twin Paradox Twins decide that one will travel to Alpha Centauri and back at 0.95c, while the other stays on earth. Compare their ages when they meet on earth. Earth twin thinks it takes 2 x 4.5 = 9 years Traveling twin thinks it takes 2 x 1.4 = 2.8 years Traveling twin will be younger! Note: Traveling twin is NOT in inertial frame!

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Question Youre eating a burger at the interstellar café 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 Always <1 L s > L M In the speeders reference frame In your reference frame

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Comparison: Time Dilation vs. Length Contraction t o = time in same reference frame as event t o = time in same reference frame as event i.e. if event is clock ticking, then t o is in the reference frame of the clock (even if the clock is in a moving spaceship). i.e. if event is clock ticking, then t o is in the reference frame of the clock (even if the clock is in a moving spaceship). L o = length in same reference frame as object L o = length in same reference frame as object length of the object when you dont think its moving. length of the object when you dont think its moving. L 0 > L Length seems shorter from outside t > t o Time seems longer from outside

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Relativistic Mass Increase Einstein made two other surprising discoveries… Einstein made two other surprising discoveries… Mass must increase with speed, as viewed by an outside observer Mass must increase with speed, as viewed by an outside observer Due to conservation of momentum Due to conservation of momentum There is leftover energy even when the object is at rest There is leftover energy even when the object is at rest Due to conservation of energy Due to conservation of energy E = mc 2 E = mc 2

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Relativistic Mass Rest mass Actually written

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E = mc 2 E = mc 2 = m 0 c 2 E = mc 2 = m 0 c 2 This E is the total energy of an object This E is the total energy of an object When the object is at rest… When the object is at rest… v = 0 v = 0 = 1 = 1 E = m 0 c 2 (rest mass energy) E = m 0 c 2 (rest mass energy) The reason that energy can be released through fusion/fission The reason that energy can be released through fusion/fission

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Total Energy Relativistic kinetic energy is the extra energy an object with mass has as a result of its motion: We can solve this for the Kinetic energy of an object:

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Relativistic Momentum Note: for v<

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Question Calculate the rest energy of an electron (m=9.1x10 -31 kg) in joules. Calculate the electrons Kinetic energy if it is moving at 0.98c.

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Simultaneous? At Rest Moving YES NO Simultaneous depends on frame! A flash of light is emitted from the exact center of a box. Does the light reach all the sides at the same time?

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Simultaneous? Many times, questions are concerned with the determination of the spatial interval and/or the time interval between two events. In this case a useful technique is to subtract from each other the appropriate Lorentz contraction describing each event.

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Three Other Effects 3 strange effects of special relativity 3 strange effects of special relativity Lorentz Transformations Lorentz Transformations Relativistic Doppler Effect Relativistic Doppler Effect Headlight Effect Headlight Effect

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Lorentz Transformations

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Light from the top of the bar has further to travel. It therefore takes longer to reach the eye. So, the bar appears bent. Weird!

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Doppler Effect The pitch of the siren: The pitch of the siren: Rises as the ambulance approaches Rises as the ambulance approaches Falls once the ambulance has passed. Falls once the ambulance has passed. The same applies to light! The same applies to light! Approaching objects appear blue (Blue-shift) Approaching objects appear blue (Blue-shift) Receding objects appear red (Red-shift) Receding objects appear red (Red-shift)

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Headlight effect Beam becomes focused. Beam becomes focused. Same amount of light concentrated in a smaller area Same amount of light concentrated in a smaller area Torch appears brighter! Torch appears brighter! V

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Warp Program used to visualise the three effects Program used to visualise the three effects

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Fun stuff Eiffel Tower Stonehenge

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Summary

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Understanding An observer has a pendulum that has a period of 3.00 seconds. His friend who happens to own a spaceship (with cool engines), zooms by the stationary pendulum. If the speedometer of the spaceship says 0.95c, what will the friend measure are the period of the pendulum? Since I am with the pendulum, my measured time is the Proper Time. This makes sense because a moving clock would run slower from my perspective. So the pendulum would have a period of 9.6s.

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Understanding Vega is 25 light-years away Vega is 25 light-years away Travel to Vega at 0.999c Travel to Vega at 0.999c The length would appear contracted to you The length would appear contracted to you About 1 light-year About 1 light-year Make the trip in ~1 light- year (each way) as measured by you Make the trip in ~1 light- year (each way) as measured by you Earth would measure 25 years each way Earth would measure 25 years each way You would spend 2 years (your time) travelling and arrive 50 years in the future Earth time. You would spend 2 years (your time) travelling and arrive 50 years in the future Earth time.

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Understanding You throw a photon (3x10 8 m/s). How fast do I think it goes when I am: You throw a photon (3x10 8 m/s). How fast do I think it goes when I am: Standing still Standing still Running 1.5x10 8 m/s towards Running 1.5x10 8 m/s towards Running 1.5x10 8 m/s away Running 1.5x10 8 m/s away Strange but True! 3x10 8 m/s

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Understanding A 1.0 m long object with a rest mass of 1.0 kg is moving at 0.90c. Find its relative length and mass Use length contraction formula: Mass increase formula:

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Understanding For a 1.0 kg mass moving at 0.90c. Find the rest energy and kinetic energy of the object For rest energy, Use energy formula: For Kinetic energy, Use relativistic energy formula: Now: Therefore

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Understanding A persons pulse rate is 65 beats per minute. a)If the person is on a spaceship moving at 0.10c, what is the pulse rate as measured by a person on Earth? b)What would the pulse rate be if the ship were moving at 0.999c? a) Use time dilation: We need time for a heart beat

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Understanding A muon at rest has an average lifespan of 2.20 x 10 -6 s a)What will an observer on Earth measure as its lifespan if it travels at 0.990c? b)What distance would we observe it travel before disintegrating? c)What distance would it travel if relativistic effects were not taken into account? a) Time dilation: b) Distance formula c) Distance formula To Show consistency This is the distance the muon measures it travels before disintegrating

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Understanding You measure the length of an object as 100m when it passes you at 0.90c. What is its length when at rest? Use length contraction formula: http://onestick.com/relativity

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Understanding As a rocket ship sweeps past the Earth with speed v, it sends out a light pulse ahead of it. How fast does the light pulse move according to the people sitting on the Earth?

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Understanding A train 0.5 km long (as measured by an observer on the train, therefore this is the proper length) is travelling at 100 km/h. Two lightening bolts strike the ends of the train simultaneously as determined by an observer on the ground. What is the time separation as measured by the observer on the train? Units: We are given that t b -t a =0 and what we want to determine is t b -t a

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Understanding A train 0.5 km long (as measured by an observer on the train, therefore this is the proper length) is travelling at 100 km/h. Two lightening bolts strike the ends of the train simultaneously as determined by an observer on the ground. What is the time separation as measured by the observer on the train? The negative sign reminds us that even a occurred after event b

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