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Date May 2019 Marks available 2 Reference code 19M.3.SL.TZ1.4
Level Standard level Paper Paper 3 Time zone 1
Command term Demonstrate Question number 4 Adapted from N/A

Question

Muons are created at a height of 3230 m above the Earth’s surface. The muons move vertically downward at a speed of 0.980 c relative to the Earth’s surface. The gamma factor for this speed is 5.00. The half-life of a muon in its rest frame is 2.20 μs.

Estimate in the Earth frame the fraction of the original muons that will reach the Earth’s surface before decaying according to Newtonian mechanics.

[3]
a.i.

Estimate in the Earth frame the fraction of the original muons that will reach the Earth’s surface before decaying according to special relativity.

[2]
a.ii.

Demonstrate how an observer moving with the same velocity as the muons accounts for the answer to (a)(ii).

[2]
b.

Markscheme

time of travel is « 3230 0.98 × 3.0 × 10 8 » = 1.10 × 10−5 «s» ✔

which is « 1.10 × 10 − 5 2.20 × 10 − 6 » = 5.0 half-lives ✔

so fraction arriving as muons is « 1 2 5 » = 1 32

OR

3 % ✔

Award [3] for a bald correct answer.

 

a.i.

time of travel corresponds to « 1.10 × 10 − 5 5.0 × 2.20 × 10 − 6 » = 1.0 half-life ✔

so fraction arriving as muons is 1 2

OR

50 % ✔

Award [2] for a bald correct answer.

 

a.ii.

observer measures the distance to the surface to be shorter «by a factor of 5.0»/ length contraction occurs ✔

so time of travel again corresponds to « ( 3230 5.0 0.98 × 3.0 × 10 8 ) ( 2.20 × 10 − 6 ) »= 1.0 half-life ✔

b.

Examiners report

Muons. The problem of muons created above the Earth surface moving almost at the speed of light, explicitly mentioned in the Guide, was solved well by prepared candidates. In i) many correctly calculated the time of travel in the earth’s frame though some struggled to recognize and apply the decay half-life aspect of this problem.

a.i.

Students who correctly answered part i) did so in ii).

a.ii.

Some candidates struggled in b), with only the best candidates identified length contraction. Importantly, the command term here is “demonstrate” which means students must make their response clear by reasoning or evidence. Many who identified length contraction did not provide adequate reasoning to gain full marks.

b.

Syllabus sections

Option A: Relativity » Option A: Relativity (Core topics) » A.2 – Lorentz transformations
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Option A: Relativity » Option A: Relativity (Core topics)
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