Special relativity — Question 176

Original QCE Vault practice · 8 marks

Q176 · Practice questionComplex unfamiliar8 marks

QUESTION 176 (8 marks)

A beam of muons moves toward the ground at

0.980c

. Their mean proper lifetime is

2.20\,\mu\mathrm s

. They are created

3.00\,\mathrm{km}

above the ground in the ground frame. Use

c=3.00\times10^8\,\mathrm{m\,s^{-1}}

and ignore changes in speed.

Determine the ground-frame mean lifetime and corresponding mean travel distance.

[3 marks]

In the muon frame, determine the distance from its creation location to the ground and the time taken for the ground to reach it.

[3 marks]

A student says that no muon can reach the ground because

vt_0=647\,\mathrm m

. Explain the errors in this argument.

[2 marks]

Question linkSyllabus coverage

WORKED SOLUTION

Practice marking scheme

8 marks

ANSWER

(a)

11.1\,\mu\mathrm s

;

3.25\,\mathrm{km}

. (b)

597\,\mathrm m

;

2.03\,\mu\mathrm s

. (c) The argument mixes a ground distance with a proper lifetime, and treats a mean lifetime as a fixed maximum.

Worked solution

(a)

\gamma=1/\sqrt{1-.980^2}=5.02519

. Thus

t=\gamma t_0=11.0554\,\mu\mathrm s

and

d=vt=.980(3.00\times10^8)(11.0554\times10^{-6})=3250.29\,\mathrm m

(b) The ground-frame

L_0=3000\,\mathrm m

is the proper separation of two locations fixed on Earth. The muon frame measures

L=L_0/\gamma=596.992\,\mathrm m

. The ground approaches at

.980c

, so

t_\mu=L/v=2.03059\,\mu\mathrm s

(c) Ground-frame distance must be compared with the dilated ground-frame lifetime; alternatively the proper lifetime must be compared with the contracted distance in the muon frame. Also muon decay is stochastic: the mean lifetime does not give a strict travel-distance cutoff for every particle.

Equivalent physically justified methods and consistent equivalent units accepted.

Displayed decimals are model answers; accept appropriate significant figures and consistent rounding from stated constants.

Part a: Calculate the Lorentz factor.

[1 mark]

Part a: Calculate the ground-frame mean lifetime.

[1 mark]

Part a: Calculate the mean travel distance.

[1 mark]

Part b: Identify the ground separation as the proper length.

[1 mark]

Part b: Apply contraction to the muon-frame distance.

[1 mark]

Part b: Calculate the muon-frame travel time.

[1 mark]

Part c: Explain the inconsistent use of quantities from different frames.

[1 mark]

Part c: Explain that a mean lifetime is not a hard upper lifetime for all muons.

[1 mark]

Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.

View the QCAA syllabus

How many marks did you earn?

Compare your working with the guide above.

Special relativity — Question 176

QUESTION 176 (8 marks)

Practice marking scheme

Related questions