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Special Relativity

Special Relativity

Special relativity starts from two postulates: physical laws are the same in every inertial frame, and the speed of light is always c for any observer. From those alone follow time dilation, length contraction, and E = mc². The effects grow as speed approaches the speed of light. Slide the speed ratio v/c here to watch time dilation increase.

Einstein's Two Postulates
💡 Why Special Relativity Was Needed
①Late 19th century: speed of light is independent of observer (Michelson–Morley experiment)
②This contradicts Newtonian mechanics! → Einstein proposed a new theory
③Postulate 1: physical laws are the same in all inertial frames (relativity)
④Postulate 2: speed of light is always c, independent of observer (invariance)
⑤From these two postulates alone, time dilation, length contraction, and E=mc² are derived!
Time Dilation — Moving Clocks Run Slow
0.6
Lorentz Factor
γ = 1√(1 - v²/c²)
v → c gives γ → ∞ (maximum effect)
Time Dilation
Δt = γΔt₀
A moving clock runs γ times slower
🔬 Muon Lifetime — Real Evidence!
①Muons: produced by cosmic rays in the upper atmosphere
②Muon lifetime ~2.2 μs → can travel only ~660 m even at the speed of light
③Yet muons formed at 10 km altitude reach the ground!
④Reason: muons travel at 0.998c, so γ ≈ 15.8
⑤Lifetime in muon frame 2.2 μs → 34.8 μs in Earth frame → can travel ~10 km!
Length Contraction — Shrinks Along Motion
Length Contraction
L = L₀γ
A moving object contracts in the direction of motion
📐 Relation Between Time Dilation and Length Contraction
①Δt₀: proper time (measured where the event occurs)
②L₀: proper length (measured at rest with the object)
③Slower time ↔ shorter length (two sides of the same effect)
④If v ≪ c, γ ≈ 1 → no noticeable difference in daily life
Mass–Energy Equivalence E = mc²
Mass–Energy Equivalence
E₀ = mc²
A mass m at rest has energy mc²
Relativistic Total Energy
E = γmc²
Total energy including kinetic = γmc²
Relativistic Kinetic Energy
Ek = (γ - 1)mc²
Total energy − rest energy = kinetic energy
💣 Energy in 1 g of Matter
①E = mc² = 0.001 × (3×10⁸)² = 9 × 10¹³ J
②Equivalent to about 21.5 kilotons of TNT (Hiroshima-scale!)
③The Sun converts ~4 million tons of mass into energy each second
④Fission/fusion: a portion of mass converts to energy
⑤'Mass + energy' conservation is the more accurate law (not just mass)
Worked Examples
Example 1
On a spaceship moving at 0.6c, a proper time of 4 s elapses. What time does a stationary Earth observer measure? (γ = 1.25)
1
Use time dilation Δt = γΔt₀.
Δt = γΔt₀
2
Substitute γ = 1.25, Δt₀ = 4 s.
Δt = 1.25 × 4 = 5 s
▸ 5 s
A moving clock runs slow (Δt > Δt₀). The proper time Δt₀ is measured in the frame where the event occurs.
Example 2
A spaceship of proper length 100 m moves at 0.8c. What length does an Earth observer measure? (γ = 5/3)
1
Use length contraction L = L₀/γ.
L = L₀γ
2
Substitute L₀ = 100 m, γ = 5/3.
L = 1005/3 = 100 × 0.6 = 60 m
▸ 60 m
Length contracts only along the motion (L < L₀). With γ = 5/3, 1/γ = 0.6, so it shrinks to 60%.
Summary
Time Dilation
Δt = γΔt₀
slowed clock
Length Contraction
L = L₀γ
shortened rod
Mass–Energy
E = mc²
Mass itself is energy
CSAT-style
How much energy is released if 2 g of rest mass is fully converted to energy? (c = 3×10⁸ m/s)
①9×10¹³ J
②1.8×10¹⁴ J
③6×10⁸ J
④1.8×10¹¹ J
⑤3.6×10¹⁴ J
▸ ② 1.8×10¹⁴ J
1
Use mass-energy equivalence E = mc² (SI: m = 0.002 kg).
E = mc2
2
Substitute m = 0.002 kg, c = 3×10⁸ m/s.
E = 0.002 × (3×108)2 = 1.8×1014 J
🎯 Exam Points
①γ = 1/√(1−v²/c²) — grows dramatically as v approaches c
②Time dilation: Δt = γΔt₀ (moving clock is slower)
③Length contraction: L = L₀/γ (only along motion direction)
④E₀ = mc² — rest energy (mass-energy equivalence)
⑤Muon lifetime, GPS corrections — real-life applications of relativity
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Matter Waves
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Chemistry · Mole Concept
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