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Date November 2014 Marks available 2 Reference code 14N.3.HL.TZ0.18
Level Higher level Paper Paper 3 Time zone Time zone 0
Command term Deduce Question number 18 Adapted from N/A

Question

This question is about muon decay.

Muons are produced in the Earth’s atmosphere at a height of around 10 km above the surface. They then travel at a speed of around 0.98c towards the Earth. The average time for a muon to decay is approximately \({\text{2.2 }}\mu {\text{s}}\), according to observers at rest relative to the muon.

Many muons are observed to reach the surface of the Earth.

Deduce that few muons would be expected to arrive at the surface of the Earth if non-relativistic mechanics are assumed.

[2]
a.

Calculate the average decay time of a muon as observed by an observer on the surface of the Earth.

[2]
b.i.

Explain, with a calculation, why many muons reach the surface of the Earth before they have decayed.

[2]
b.ii.

Markscheme

\(t = \left( {\frac{d}{v} = \frac{{{{10}^4}}}{{0.98 \times 3 \times {{10}^8}}} = } \right){\text{ }}3.4 \times {10^{ - 5}}{\text{ s}}\);

which is \(\frac{{3.4 \times {{10}^{ - 5}}}}{{2.2 \times {{10}^{ - 6}}}} \approx 15\) decay times;

(so very few muons will reach Earth)

or

\(d = vt = 0.98 \times 3 \times {10^8} \times 2.2 \times {10^{ - 6}} = 650{\text{ m}}\) in one decay time;

in travelling to Earth, there are \(\frac{{10000}}{{650}} \approx 15\) decay times;

(so very few muons will reach Earth)

a.

\(\gamma  = \sqrt {\frac{1}{{1 - {{0.98}^2}}}}  = 5.0\);

\(t = \gamma {t_0} = 5.0 \times 2.2 \times {10^{ - 6}} = 1.1 \times {10^{ - 5}}{\text{ s}}\);

b.i.

distance travelled in one half-life is

\(d = vt = 0.98 \times 3 \times {10^8} \times 1.1 \times {10^{ - 5}} = 3200{\text{ m}}\);

for observers on Earth, the muons are able to travel much further compared with those in (a);

(so many more will be able to reach the Earth before they have decayed)

b.ii.

Examiners report

Many well prepared candidates presented good, well structured and clear answers.

a.

Many well prepared candidates presented good, well structured and clear answers.

b.i.

Many well prepared candidates presented good, well structured and clear answers. Some candidates struggled with (b)(ii).

b.ii.

Syllabus sections

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