IB Diploma Physics · topic A.5
Galilean and special relativity
Connect the ideas, make your working visible, and test the result. Use the hints when you need a starting point; open the full solution after your attempt.
Extra challenging questions
Work through the linked stages and explain what each result means.
A short-lived particle reaches farther than a classical prediction
A particle's proper mean lifetime is 2.0 μs. It travels at 0.80c relative to a laboratory. Use c = 3.0×10⁸ m s⁻¹ and γ = 1/√(1−v²/c²). The graph supplies γ for selected speeds. The particle's rest energy is 100 MeV. Supplied data (Speed / c; γ): (0, 1), (0.6, 1.25), (0.8, 1.6666667), (0.9, 2.2941573). (a) Find the laboratory mean lifetime, mean travel distance and kinetic energy. [3 marks] (b) Explain the same survival-distance result in the particle rest frame and why Galilean velocity addition fails for light. [3 marks]
Lorentz factor
Supplied question data. Read the labelled axes and use the observations to test the physical model.
Vertical axis (y): γ
Horizontal axis (x): Speed / c
Read the exact data as a table
| Speed / c | γ |
|---|---|
| 0 | 1 |
| 0.6 | 1.25 |
| 0.8 | 1.6666667 |
| 0.9 | 2.2941573 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Begin with part (a): Find the laboratory mean lifetime, mean travel distance and kinetic energy.
Reveal the worked solution
- At 0.80c, γ = 1/√0.36 = 5/3. The proper lifetime is measured in the particle rest frame, so laboratory lifetime is γτ0 = 3.33 μs.
- Laboratory mean distance is vτlab = 0.80(3.0×10⁸)(3.33×10⁻⁶) = 800 m.
- Relativistic kinetic energy is (γ−1)mc² = (2/3)(100)≈66.7 MeV. Using ½mv² at this speed gives an inaccurate result.
- The relevant 800 m laboratory path is parallel to relative motion, so the particle frame measures 800/γ = 480 m.
- The laboratory and detector move toward the particle at 0.80c; their approach time is 480/(0.80c) = 2.0 μs, consistent with the proper lifetime.
- Galilean addition would give different light speeds in different inertial frames. Special relativity uses Lorentz transformations and invariant vacuum light speed; time dilation and length contraction are linked features of that model.
Answer (a) Lab lifetime≈3.33 μs; mean travel distance=800 m; kinetic energy≈66.7 MeV. (b) The laboratory distance is length-contracted to 480 m in the particle frame, traversed in the 2.0 μs proper mean lifetime.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.