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Grade 11-12 (age 16-18)

high school physics Special Relativity

Special relativity starts from two postulates: the laws of physics are the same in every inertial frame, and the speed of light is c for every observer. From these two postulates alone follow time dilation, in which moving clocks run slow, length contraction, in which moving objects are shortened, and E = mc², in which mass is energy. These effects grow as the speed approaches the speed of light and are almost invisible at everyday speeds. In the pictures, changing the speed ratio v/c changes the two clocks and the two rods together, and both are set by the same Lorentz factor γ.

Einstein's Two Postulates

💡 Why Was Special Relativity Needed?
①In the late 19th century, the Michelson–Morley experiment found that the speed of light did not change even when the observer moved
②This result did not fit the Galilean rule of simply adding velocities, and Einstein proposed a new theory
③Postulate 1 (principle of relativity): the laws of physics are the same in all inertial frames
④Postulate 2 (constancy of the speed of light): the speed of light is always c, whatever the observer
⑤Time dilation, length contraction and E = mc² all follow from these two postulates

Time Dilation: Moving Clocks Run Slow

0.60
Lorentz Factor
γ = 1√(1 - v²/c²)
As v approaches c, γ grows without limit
Time Dilation
Δt = γΔt₀
The time Δt measured by an observer at rest is γ times the proper time Δt₀ of the moving clock
🔬 The Lifetime of Muons: Real Evidence
①Muons are particles made high in the atmosphere by high-energy particles arriving from space
②A muon lives about 2.2 μs, so even at the speed of light it could travel only about 660 m
③Yet muons made about 10 km up are detected at the ground
④The reason is that they arrive at 0.998c, so γ ≈ 15.8
⑤The muon's proper lifetime of 2.2 μs stretches to about 34.8 μs as seen from Earth, enough to travel about 10 km

Length Contraction: Shorter Along the Motion

Length Contraction
L = L₀γ
A moving object is shortened only along the direction of motion, to 1/γ of its proper length
📐 Time Dilation and Length Contraction Go Together
①Δt₀ is the proper time, measured by a clock moving with the event
②L₀ is the proper length, measured while at rest with the object
③With the same γ, time stretches and length shrinks; they are the same effect seen from two directions
④When v is much smaller than c, γ ≈ 1, so no difference can be felt in daily life

Mass–Energy Equivalence E = mc²

Mass–Energy Equivalence
E₀ = mc²
A mass m at rest has an energy mc² simply by being mass
Relativistic Total Energy
E = γmc²
The total energy, kinetic energy included, is γmc²
Relativistic Kinetic Energy
Ek = (γ - 1)mc²
Total energy − rest energy = kinetic energy
💣 The Energy in 1 g
①The rest energy of 1 g of mass is E = mc² = 0.001 × (3 × 10⁸)² = 9 × 10¹³ J
②That is enough to keep a 100 W light bulb on for about 28,000 years
③Every second the Sun turns about 4 million tonnes of mass into energy as light and heat
④Nuclear fission and fusion are reactions in which part of the mass becomes energy
⑤So conservation of mass and energy counted together is a more accurate law than conservation of mass alone

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 and Δt₀ = 4 s.
Δt = 1.25 × 4 = 5 s
▸ 5 s
While 4 s pass on the spaceship, 5 s pass on Earth. First decide which is the proper time Δt₀: the time on a clock moving with the event.
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 and γ = 5/3.
L = 1005/3 = 100 × 0.6 = 60 m
▸ 60 m
Since 1/γ = 0.6, the length shrinks to 60% of the proper length, and only the length along the motion shrinks.

Summary

Time Dilation
Δt = γΔt₀
Moving clocks run slow
Length Contraction
L = L₀γ
Moving rods are shorter
Mass–Energy
E = mc²
Mass itself is energy
CSAT-style problem, adapted
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, converting the mass to SI units: m = 0.002 kg.
E = mc2
2
Substitute m = 0.002 kg and c = 3 × 10⁸ m/s.
E = 0.002 × (3×108)2 = 1.8×1014 J
🎯 Exam Points
①γ = 1/√(1 − v²/c²) grows sharply as v approaches c
②Time dilation: Δt = γΔt₀ (moving clocks run slow)
③Length contraction: L = L₀/γ (only along the direction of motion)
④The rest energy E₀ = mc² means mass is energy
⑤Muons reaching the ground are a classic piece of evidence for time dilation
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