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Special Relativity Practice

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Special Relativity Practice

A particle detector has recorded fast-moving muons, and your research team needs to check its relativity calculations. Use c = 3.00 × 10⁸ m s⁻¹. Give answers to 3 significant figures where appropriate. Show your working for calculations.

Frames, postulates and evidence

Build a sound foundation before analysing the detector results.

1.A laboratory frame moves at constant velocity relative to a distant star, with no rotation or acceleration. Explain why it can be treated as an inertial frame, and state what would make a frame non-inertial.

2.A trainee writes: “The speed of light is c only for an observer at rest in the laboratory.” Which option correctly states Einstein’s two postulates and identifies the error?

The laws of physics are the same in all inertial frames, and light in vacuum has the same speed c in all inertial frames; the trainee wrongly assumes a preferred rest frame.
The laws of physics are the same in all accelerating frames, and light’s speed depends on the source; the trainee is correct.
Energy is conserved in every frame, and all observers measure the same time; the trainee confuses energy with speed.
The laws of physics apply only in the laboratory, and light has speed c relative to its source; the trainee is correct.

3.Select every observation that supports special relativity.

The Michelson–Morley experiment found no expected change in the measured speed of light due to Earth’s motion.
Fast muons are detected at Earth’s surface in greater numbers than a classical lifetime estimate predicts.
Satellite navigation systems require relativistic timing corrections.
Classical Galilean velocity addition accurately predicts light’s speed for observers moving near c.

4.Match each term to its definition. Write the matching number beside each letter.

A. Proper time, Δt₀
B. Dilated time, Δt
C. Proper length, L₀
D. Lorentz factor, γ

2. The time interval between events occurring at the same place in the clock’s rest frame
4. The time interval measured by an observer for a moving clock; it is longer than proper time
3. The factor γ = 1/√(1 − v²/c²)
1. The length of an object measured in its rest frame

Calculate and interpret

Use γ = 1/√(1 − v²/c²), Δt = γΔt₀ and L = L₀/γ for motion along the measured length.

5.A muon has a proper lifetime of 2.20 μs. It travels at 0.980c relative to a detector on Earth. Calculate γ and the muon’s lifetime measured by the detector.

6.A probe’s rest length is 10.0 m. An observer measures it moving parallel to its length at 0.600c. Calculate the length measured by that observer.

7.A particle has rest mass m₀. State its rest energy and write an expression for its total relativistic energy when it moves at speed v. Explain briefly how these expressions are related.

8.In Bertozzi’s electron experiment, measurements of electron speed and kinetic energy showed that adding energy made the electrons approach c without reaching it. Explain why this challenges the classical formula Eₖ = ½m₀v² and how relativity accounts for the result.

The speed limit

Use the Lorentz factor to explain what happens at the limit.

9.A student claims that a spacecraft with non-zero rest mass can be accelerated to exactly c if its engine runs long enough. Give one concise reason, using the behaviour of γ, why this claim is incorrect.