Special relativity — Question 180

Original QCE Vault practice · 6 marks

Q180 · Practice questionComplex unfamiliar6 marks

QUESTION 180 (6 marks)

An accelerator measures the following momentum magnitudes for the same particle. Use

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

$v/c$	0.600	0.800	0.900
$p$ ( $10^{-19}\,\mathrm{kg\,m\,s^{-1}}$ )	4.50	8.00	12.4

Use numerical evidence to show why a constant-mass classical relation

p=mv

cannot fit all three results.

[2 marks]

Determine the rest mass using the

0.600c

result.

[2 marks]

Use this rest mass to predict the momentum at

0.900c

and compare with the measurement.

[2 marks]

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WORKED SOLUTION

Practice marking scheme

6 marks

ANSWER

(a)

p/v

rises from

2.50

3.33

4.59

, in units of

10^{-27}\,\mathrm{kg}

. (b)

m_0=2.00\times10^{-27}\,\mathrm{kg}

. (c)

1.24\times10^{-18}\,\mathrm{kg\,m\,s^{-1}}

, consistent with the measured

12.4\times10^{-19}

Worked solution

(a) The implied classical masses are

4.50\times10^{-19}/(.600c)=2.50\times10^{-27}

8.00\times10^{-19}/(.800c)=3.33\times10^{-27}

and

12.4\times10^{-19}/(.900c)=4.59\times10^{-27}\,\mathrm{kg}

. Since

p/v

is not constant, one fixed classical mass does not fit.

(b)

m_0=p\sqrt{1-v^2/c^2}/v=(4.50\times10^{-19})(.800)/(.600c)=2.00\times10^{-27}\,\mathrm{kg}

(c)

p=m_0(.900c)/\sqrt{1-.900^2}=1.23884\times10^{-18}\,\mathrm{kg\,m\,s^{-1}}

. This rounds to the measured value, supporting the relativistic model to the precision supplied.

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 at least two different values of p divided by v.

[1 mark]

Part a: Explain why their difference rejects a fixed-mass classical proportionality.

[1 mark]

Part b: Rearrange the relativistic momentum relation.

[1 mark]

Part b: Calculate rest mass.

[1 mark]

Part c: Calculate the relativistic prediction.

[1 mark]

Part c: Compare with the observed value using its rounding precision.

[1 mark]

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

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Special relativity — Question 180

QUESTION 180 (6 marks)

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