| Date | May 2013 | Marks available | 2 | Reference code | 13M.3.HL.TZ2.16 |
| Level | Higher level | Paper | Paper 3 | Time zone | Time zone 2 |
| Command term | Show | Question number | 16 | Adapted from | N/A |
Question
This question is about muon decay experiments.
Muons created high in the atmosphere move vertically down towards the surface of Earth. A muon detector T is placed on top of a mountain and another, detector S, is placed at sea level.
Detector T detects 570 muons per hour. In the rest frame of the muons their half-life is 1.5 μs. According to an observer, at rest on the mountain, the muons take 6.0 μs to travel from detector T to detector S.
Show that, if the muons move at non-relativistic speed, the number of muons detected at sea level would be approximately 36 per hour.
The muons in (a) move toward the surface of Earth with a relativistic speed of 0.968c.
(i) Determine the half-life of the muons according to the observer at rest on the mountain.
(ii) The number of muons observed at detector S is 285 per hour. Explain, using your answers to (a) and (b)(i), how this observation provides evidence for time dilation.
Markscheme
the number of half-lives that go by until muons make it to sea level is \(\frac{{6.0}}{{1.5}} = 4\);
and so the number of muons per hour would be \(\frac{{570}}{{{2^4}}} = \frac{{570}}{{16}}\left( { = 35.6} \right)\);
(≈36)
Answer given, reward correct working only.
(i) y= 3.985;
so the dilated half-life is 3.985×1.5 = 5.977 ≈ 6.0(μs);
(ii) 285 muons per hour represents a half-life of 6.0(μs);
this time is four times greater than in the muon frame, and so confirms time dilation;
Allow similar arguments explaining that the two different half-lives or count-rates are proof of time dilation.
Examiners report
(a) provided an easy two marks for the majority, as did (b)(i) for the muon half-life in the Earth rest-frame.
(a) provided an easy two marks for the majority, as did (b)(i) for the muon half-life in the Earth rest-frame. In (b)(ii) not all candidates referred to their previous answers, as directed, in explaining the evidence for time dilation. There is uncertainty about the meaning of the words 'dilated time' and who perceives it. Candidates need to know that dilated means larger/longer and refers to the longer elapsed time on two different clocks in the observers frame compared to the shorter elapsed 'proper time' on a single, relatively moving clock. In other words, the relatively moving clock's elapsed time (between two events at which this clock is present) is always less than that between the observer's two clocks separated in space. This is a very difficult concept and candidates really need to try to get to grips with it. Without reference to clocks it is almost impossible to understand time dilation or simultaneity. Spacetime diagrams are helpful again here. Due to this misunderstanding there were many references to 'time running slow for the muons'. This is incorrect. Muon's time runs slow for the Earth observers. The difference is subtle but important.
