Twin Paradox The Theory of Relativity. About Relativity As an object approaches the speed of light,...
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Transcript of Twin Paradox The Theory of Relativity. About Relativity As an object approaches the speed of light,...
![Page 1: Twin Paradox The Theory of Relativity. About Relativity As an object approaches the speed of light, time slows down. (Moving clocks are slow) (Moving.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e9f5503460f94ba1093/html5/thumbnails/1.jpg)
Twin Paradox
The Theory of Relativity
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About Relativity
• As an object approaches the speed of light, time slows down.
• (Moving clocks are slow)
• (Moving rulers are short)
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A trip from A trip from EarthEarth to to PlanetPlanet HollywoodHollywood
►Homer stays on Earth.Homer stays on Earth.►Loner travels 10 light-years at 80% of the Loner travels 10 light-years at 80% of the
speed of light. (speed of light = c)speed of light. (speed of light = c)►Beta (Beta (ββ) is the velocity of the object ) is the velocity of the object
compared to the speed of light. (compared to the speed of light. (ββ = 4/5c = = 4/5c = 0.8c) 0.8c)
►Gamma (Gamma (γγ) is the effect of traveling at ) is the effect of traveling at speeds close to light speed (c) has on time speeds close to light speed (c) has on time (t) or distance (x).(t) or distance (x).
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Earth
Homer
Loner
Planet Hollywo
od
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Effect of Time on Spaceship
• Velocity (v) is v = 4/5c = 0.8c therefore…
• β = 4/5 = 0.8
γ = 1
1 - β²1 - β² = 1 - 0.64 = 0.36 and the √ of 0.36 is 0.6 !!
(β² = 0.8² = 0.64)
γ = 1 ÷ 0.6 = 1²/³ = 5/3
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As Viewed From EarthAs Viewed From Earth
Without Relativity…..Without Relativity….. xx = = vvt t or or tt = = xx//vv x is light years traveledx is light years traveled v is velocity.v is velocity. t is time.t is time. t t = = 1010//0.8c0.8c = 12.5 years each way. = 12.5 years each way. There and back makes the trip There and back makes the trip
12.5 x 2 or 25 years!!12.5 x 2 or 25 years!!
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As Viewed From Earth As Viewed From Earth
•With RelativityWith Relativity• Homer sees Loner’s Homer sees Loner’s
clock is running slow clock is running slow by…by…
• γγ = 5/3 !! = 5/3 !!
• Therefore Loner’s clock Therefore Loner’s clock reads 25 years ÷ reads 25 years ÷ γγ
• 25 ÷25 ÷ 5/3 = 5/3 = 15 years!!15 years!!
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Physical Results of Trip
Homer on Earth ages 25 years!!
Loner, traveling at 80% the speed of light ages 15 years!!
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As Viewed From Spaceship• Loner sees distance of planets contracted by
γ = 5/3
• In Loner’s frame distance is 10 light years ÷ 5/3 10 ÷ 5/3 = 6 light years.
• Therefore t = x/v = 6/0.8 = 7.5 years each way.
• There and back is 7.5 x 2 = 15 year trip for Loner!!
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Cosmic MuonsCosmic MuonsFactor:Factor:
γ =1
1 - v²/c² (β² = v²/c²)
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Muon LifetimeMuon Lifetime
d = d = vvtt t = t = 2 x 10(-6)2 x 10(-6) seconds secondsv = v = 3 x 10(8)3 x 10(8) m/s m/sd = d = 2 x 10(-6)2 x 10(-6) • • 3 x 10(8)3 x 10(8) = 600m = 0.6km = 600m = 0.6km
A muon only travels 0.6 km A muon only travels 0.6 km into the atmosphere!into the atmosphere!
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Question??Question??
• Muons are created in the upper atmosphere.
• Detectors tell us they reach the Earth’s surface.
• Suppose the distance needed to travel is approximately 10 km.
• If muons travel only 0.6 km, how do they reach the Earth’s surface??
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With Relativity suppose v = 0.999c
γ =1
1 - v²/c² = 22
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Earth’s Reference FrameEarth’s Reference Frame
• Muon’s clock runs slow by γ.
• Therefore the muon lives 22x longer and travels 22x further!
• So… d = (0.6 km) (22) = 13.2 km
• 13.2 km > 10.0 km allowing for the muon to reach the Earth’s surface!!
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Earth’s Reference FrameEarth’s Reference Frame
Muon heading towards Earth at
0.999c
Homer notices that the muon’s “clock” is running slow
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Muon’s Reference FrameMuon’s Reference Frame
Earth’s atmosphere moves towards the Earth’s atmosphere moves towards the muon at v = 0.999c, therefore the muon at v = 0.999c, therefore the atmosphere is contracted by atmosphere is contracted by γγ..
Thickness of atmosphere is 10km/Thickness of atmosphere is 10km/22 22 or or 0.45 km to the muon.0.45 km to the muon.
The muon travels 0.6 km and 0.6 > 0.45 The muon travels 0.6 km and 0.6 > 0.45 so the muon reaches Earth!!so the muon reaches Earth!!
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Muon’s Reference Frame
Earth
Muon’s atmosphere 0.45km
Astronaut’s atmosphere 10km
Traveling at 0.999cThe muon’s ruler is “shortened,” while The astronauts, barelyMoving in comparison, See a thicker atmosphere.
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The End!!