Twin Paradox Theory of Relativity About Relativity As
Twin Paradox Theory of Relativity
About Relativity • As an object approaches the speed of light, time slows down. • (Moving clocks are slow) • (Moving rulers are short)
A trip from Earth to Planet Hollywood ► Homer stays on Earth. ► Loner travels 10 light-years at 80% of the speed of light. (speed of light = c) ► Beta (β) is the velocity of the object compared to the speed of light. (β = 4/5 c = 0. 8 c) ► Gamma (γ) is the effect of traveling at speeds close to light speed (c) has on time (t) or distance (x).
Planet Hollywood Loner Home r Earth
Effect of Time on Spaceship • Velocity (v) is v = 4/5 c = 0. 8 c therefore… • β = 4/5 = 0. 8 γ= 1 1 - β² (β² = 0. 8² = 0. 64) 1 - β² = 1 - 0. 64 = 0. 36 and the √ of 0. 36 is 0. 6 !! γ = 1 ÷ 0. 6 = 1²/³ = 5/3
As Viewed From Earth n Without Relativity…. . x = vt or t = x/v n x is light years traveled n v is velocity. n t is time. n t = 10/0. 8 c = 12. 5 years each way. n There and back makes the trip 12. 5 x 2 or 25 years!! n
As Viewed From Earth • With Relativity • Homer sees Loner’s clock is • • • running slow by… γ = 5/3 !! Therefore Loner’s clock reads 25 years ÷ γ 25 ÷ 5/3 = 15 years!!
Physical Results of Trip n Homer on Earth ages 25 years!! n Loner, traveling at 80% the speed of light ages 15 years!!
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!!
Cosmic Muons Factor: γ= 1 1 - v²/c² (β² = v²/c²)
Muon Lifetime d = vt l t = 2 x 10(-6) seconds l v = 3 x 10(8) m/s l d = 2 x 10(-6) • 3 x 10(8) = 600 m = 0. 6 km l l. A muon only travels 0. 6 km into the atmosphere!
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? ?
With Relativity n suppose v = 0. 999 c γ= 1 1 - v²/c² = 22
Earth’s Reference Frame • Muon’s clock runs slow by γ. • Therefore the muon lives 22 x longer and travels 22 x 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!!
Earth’s Reference Frame Muon heading towards Earth at 0. 999 c Homer notices that the muon’s “clock” is running slow
Muon’s Reference Frame Earth’s atmosphere moves towards the muon at v = 0. 999 c, therefore the atmosphere is contracted by γ. Thickness of atmosphere is 10 km/22 or 0. 45 km to the muon. The muon travels 0. 6 km and 0. 6 > 0. 45 so the muon reaches Earth!!
Muon’s Reference Frame Traveling at 0. 999 c The muon’s ruler is “shortened, ” while The astronauts, barely Moving in comparison, See a thicker atmosphere. Muon’s atmosphere 0. 45 km Earth Astronaut’s atmosphere 10 km
The End!!
- Slides: 18