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PHYSICS, M7 2025 HSC 32

Analyse the consequences of the theory of special relativity in relation to length, time and motion. Support your answer with reference to experimental evidence.   (8 marks)

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Overview Statement

  • Special relativity predicts that time dilation, length contraction and relativistic momentum arise from the principle that the speed of light is constant for all observers.
  • These effects change how time, distance and motion are measured and each consequence is supported by experimental evidence.

Time–Length Relationship in Muon Observations

  • Muon-decay experiments show how time dilation and length contraction depend on relative motion.
  • Muons created high in the atmosphere have such short lifetimes that, in classical physics, they should decay long before reaching Earth’s surface. Yet far more muons are detected at ground level than predicted.
  • In Earth’s frame of reference, the muons experience time dilation, so they “live longer” and travel further.
  • In the muon’s frame of reference, the atmosphere is length-contracted according to the equation  \(l=l_0 \sqrt{\left(1-\dfrac{v^{2}}{c^{2}}\right)}\), so the distance they travel is much shorter.
  • Both viewpoints are valid within their own reference frames, showing that time and length are not absolute but depend on relative motion.

Momentum–Energy Relationship in Particle Accelerators

  • Relativistic momentum explains how objects behave as they approach light speed.
  • Particle accelerators show that enormous increases in energy produce only small increases in speed at high velocities.
  • Particles act as though their mass increases, so each additional acceleration requires disproportionately more energy.
  • This makes it impossible for any object with mass to reach the speed of light, since doing so requires infinite energy.
  • Thus, relativistic momentum preserves light speed as a universal limit.

Implications and Synthesis

  • The consequences of these observations reveal that space and time form a single, interconnected framework rather than separate absolute quantities.
  • At high velocities, motion fundamentally alters measurements of time, distance and momentum.
  • Together, these consequences confirm that classical physics fails at relativistic speeds and that special relativity accurately describes the behaviour of fast-moving objects.
Show Worked Solution

Overview Statement

  • Special relativity predicts that time dilation, length contraction and relativistic momentum arise from the principle that the speed of light is constant for all observers.
  • These effects change how time, distance and motion are measured and each consequence is supported by experimental evidence.

Time–Length Relationship in Muon Observations

  • Muon-decay experiments show how time dilation and length contraction depend on relative motion.
  • Muons created high in the atmosphere have such short lifetimes that, in classical physics, they should decay long before reaching Earth’s surface. Yet far more muons are detected at ground level than predicted.
  • In Earth’s frame of reference, the muons experience time dilation, so they “live longer” and travel further.
  • In the muon’s frame of reference, the atmosphere is length-contracted according to the equation  \(l=l_0 \sqrt{\left(1-\dfrac{v^{2}}{c^{2}}\right)}\), so the distance they travel is much shorter.
  • Both viewpoints are valid within their own reference frames, showing that time and length are not absolute but depend on relative motion.

Momentum–Energy Relationship in Particle Accelerators

  • Relativistic momentum explains how objects behave as they approach light speed.
  • Particle accelerators show that enormous increases in energy produce only small increases in speed at high velocities.
  • Particles act as though their mass increases, so each additional acceleration requires disproportionately more energy.
  • This makes it impossible for any object with mass to reach the speed of light, since doing so requires infinite energy.
  • Thus, relativistic momentum preserves light speed as a universal limit.

Implications and Synthesis

  • The consequences of these observations reveal that space and time form a single, interconnected framework rather than separate absolute quantities.
  • At high velocities, motion fundamentally alters measurements of time, distance and momentum.
  • Together, these consequences confirm that classical physics fails at relativistic speeds and that special relativity accurately describes the behaviour of fast-moving objects.

PHYSICS, M7 2024 HSC 26

Muons are unstable particles produced when cosmic rays strike atoms high in the atmosphere. The muons travel downward, perpendicular to Earth's surface, at almost the speed of light.

Classical physics predicts that these muons will decay before they have time to reach Earth's surface.

Explain qualitatively why these muons can reach Earth's surface, regardless of whether their motion is considered from either the muon's frame of reference or the Earth's frame of reference.   (3 marks)

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  • The muon’s are able to reach the Earth’s surface due to Einstein’s special theory of relativity in relation to length contraction and time dilation.

Muon’s frame of reference:

  • The distance to the Earth’s surface is contracted according to  \(l=l_o\sqrt{1-\frac{v^2}{c^2}}\). Muons see the Earth’s surface move towards them at speeds close to \(c\).
  • Since the muons have to travel a shorter distance than the proper length, they will have time to reach the Earth’s surface before they decay. 

Earth’s frame of reference:

  • The time that it takes the muon to decay will be dilated according to  \(t=\dfrac{t_o}{\sqrt{1-\frac{v^2}{c^2}}}\)  as the muon’s are moving close to the speed of light.
  • Therefore, the muons have a longer half-life and lifespan than predicted by classical physics and will be able to reach the Earth’s surface before they decay.
  • In this way, muons can reach the surface of the Earth from either frame of reference. 

Show Worked Solution

  • The muon’s are able to reach the Earth’s surface due to Einstein’s special theory of relativity in relation to length contraction and time dilation.

Muon’s frame of reference:

  • The distance to the Earth’s surface is contracted according to  \(l=l_o\sqrt{1-\frac{v^2}{c^2}}\). Muons see the Earth’s surface move towards them at speeds close to \(c\).
  • Since the muons have to travel a shorter distance than the proper length, they will have time to reach the Earth’s surface before they decay.

Earth’s frame of reference:

  • The time that it takes the muon to decay will be dilated according to  \(t=\dfrac{t_o}{\sqrt{1-\frac{v^2}{c^2}}}\)  as the muon’s are moving close to the speed of light.
  • Therefore, the muons have a longer half-life and lifespan than predicted by classical physics and will be able to reach the Earth’s surface before they decay.
  • In this way, muons can reach the surface of the Earth from either frame of reference. 
♦ Mean mark 46%.

PHYSICS, M7 2024 HSC 20 MC

Three identical atomic clocks are made so that they tick at precisely the same rate. One is kept in a laboratory, \(X\), on Earth's equator. Another is placed on board a satellite, \(Y\), in a circular orbit with a period of 12 hours. A third is placed in a satellite, \(Z\), that is in a geostationary orbit. The satellites orbit Earth in the equatorial plane.
 

Assume that the satellites are inertial frames of reference and the clocks are affected ONLY by the predictions of special relativity.

Which statement correctly compares the rates at which the clocks tick, as determined by an observer at \(X\), when the satellites are in the positions shown in the diagram?

  1. The clock at \(Y\) ticks faster than either the clock at \(X\) or the clock at \(Z\).
  2. The clock at \(Y\) ticks slower than either the clock at \(X\) or the clock at \(Z\).
  3. The clocks tick at different rates, with \(X\) being the fastest and \(Y\) being the slowest.
  4. The clocks tick at different rates, with \(Z\) being the slowest and \(X\) being the fastest.
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\(B\)

Show Worked Solution
  • Using Einstein’s special theory of relativity in relation to time dilation, the faster a clock travels relative to a stationary observer, the slower time moves for the clock. This means that the clock moving relative to the observer will tick more slowly.
  • The clock placed at \(X\) will be stationary relative to the observer at \(X\).
  • This is also true for the clock placed at \(Z\). As the clock being placed in a satellite which is in a geostationary orbit, the satellite will appear to be stationary in the sky. Therefore, the observer at \(X\) is in the same frame of reference as the clock at \(Z\) and no effects of time dilation will be observed.
  • The clock at \(Y\) has a period of 12 hours, hence it must have a smaller orbital radius and so a higher linear velocity than \(Z\). Thus the clock at \(Y\) would be moving faster as seen by the observer at \(X\) which is in the same frame of reference as \(Z\). Therefore, the observer at \(X\) would see the clock at \(Y\) tick slower than the clocks at \(X\) and \(Z\) due to the effects of time dilation.

\(\Rightarrow B\)

Note: As the question assumes that all satellites are in inertial frames of reference, students can discount the rotational velocities of the satellites and the centripetal forces of gravity on the satellites, effectively treating the Earth as flat.

♦♦♦ Mean mark 21%.

PHYSICS, M7 2020 VCE 11

An astronaut has left Earth and is travelling on a spaceship at 0.800\(c\) directly towards the star known as Sirius, which is located 8.61 light-years away from Earth, as measured by observers on Earth.

  1. How long will the trip take according to a clock that the astronaut is carrying on his spaceship? Show your working.   (2 marks)
  1. Is the trip time measured by the astronaut in part (a) a proper time? Explain your reasoning.   (2 marks)

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a.    \(6.46\ \text{years}\)

b.    Proper time measurement

  • The trip time of 6.46 years on the spaceship is a proper time.
  • This is due to the Astronaut’s clock being stationary within the astronaut’s frame of reference.

Show Worked Solution

a.    From the earth’s perspective:

\(\text{Travel time}\ =\dfrac{8.61}{0.8}=10.76\ \text{years}\).

From the astronaut’s perspective:

  • The Earth’s time is going to be dilated compared to his, so the time the astronauts clock will measure is:

\(t=\dfrac{t_0}{\sqrt{1-\frac{v^2}{c^2}}}\)

\(t_0\) \(=t\sqrt{1-\frac{v^2}{c^2}}\)
  \(=10.76 \times \sqrt{1-\frac{(0.8c)^2}{c^2}}\)
  \(=10.76 \times \sqrt{1-0.8^2}\)
  \(=6.46\ \text{years}\)

♦♦♦ Mean mark (a) 29%.
COMMENT: Light years is a measure of distance, not time!

b.    Proper time measurement

  • The trip time of 6.46 years on the spaceship is a proper time.
  • This is due to the Astronaut’s clock being stationary within the astronaut’s frame of reference.
♦ Mean mark (b) 44%.

PHYSICS, M7 2022 VCE 11

Explain why muons formed in the outer atmosphere can reach the surface of Earth even though their half-lives indicate that they should decay well before reaching Earth's surface.    (2 marks)

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Using the special relativity principal of time dilation:

  • The average lifespans of muons are 2.2 μs, but when they are observed from the Earth’s frame of reference this becomes significantly dilated.
  • Thus, they are able to travel from the outer atmosphere and reach the surface of the Earth before they decay.
  • Other answers could have also referred to length contraction from the perspective of the muons.
Show Worked Solution

Answer 1: Using the special relativity principal of time dilation:

  • The average lifespans of muons are 2.2 μs, but when they are observed from the Earth’s frame of reference this becomes significantly dilated.
  • Thus, they are able to travel from the outer atmosphere and reach the surface of the Earth before they decay.
  • Other answers could have also referred to length contraction from the perspective of the muons.

Answer 2: Using the special relativity principal of length contraction:

  • A muon’s frame of reference measures the distance to the Earth as shorter than that measured from the Earth’s frame of reference.
  • This shorter distance allows the muons to reach the Earth before they decay.
♦ Mean mark 41%.

PHYSICS, M7 2023 HSC 22

A spacecraft passes Earth at a speed of 0.9\(c\). The spacecraft emits a light pulse every 3.1 \(\times\) 10\(^{-9}\) s, as measured by the crew on the spacecraft.

What is the time between the pulses, as measured by an observer on Earth?  (3 marks)

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\(7.1 \times 10^{-9}\ \text{s} \)

Show Worked Solution

\(t\) \(=\dfrac{t_o}{\sqrt{(1- \frac{v^2}{c^2})}} \)  
  \(=\dfrac{3.1 \times 10^{-9}}{\sqrt{(1-\frac{(0.9c)^2}{c^2})}}\)  
  \( =\dfrac{3.1 \times 10{-9}}{\sqrt{(1-{0.9}^2)}} \)  
  \(=7.1 \times 10^{-9}\ \text{s} \)  

PHYSICS, M7 EQ-Bank 16 MC

In 1972 , four caesium clocks were flown twice around the world on commercial jet flights, once eastward and once westward. The travelling clocks were compared with reference clocks at the US Naval Observatory and the results were compared with predictions from Einstein's theory of special relativity.
 

Which of the following is correct about the observed results in relation to Einstein's theory?

  1. Both of the results are inconclusive.
  2. Both of the results support the theory.
  3. One of the results supports the theory and the other is inconclusive.
  4. One of the results supports the theory and the other rejects the theory.
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`B`

Show Worked Solution
  • The tolerance range for the eastward journey is –63 to –17 nanoseconds. 
  • The tolerance range for the westward journey is 254 to 296 nanoseconds. 
  • Therefore, both of the results fall within the range predicted by Einstein’s theory of special relativity.

`=>B`

PHYSICS, M7 2016 HSC 19 MC

Muons are subatomic particles which at rest have a lifetime of 2.2 microseconds `(mus)`. When they are produced in Earth's upper atmosphere, they travel at 0.9999 `c`.

Using classical physics, the distance travelled by a muon in its lifetime can be calculated as follows:

`x=vt=660\ text{m}`

Which row of the table correctly summarises the behaviour of these muons?
 

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`A`

Show Worked Solution
  • In the frame of reference of the muon, classical physics applies.
  • Hence, the muon will experience its proper, or actual lifespan and distance travelled.
  • In the frame of reference of an observer on earth, due to the muons relativistic speed relative to the observer, time dilation of the muons lifespan occurs.
  • The lifetime of a muon will be greater than 2.2 `mus` from Earth’s frame of reference.

`=>A`


♦ Mean mark 49%.

PHYSICS, M7 2017 HSC 23

Using examples from special relativity, explain how theories in science are validated in different ways.   (5 marks)

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  • Theories in science must be consistent with observations and supported by independent, valid experiments in order to be validated.
  • Special relativity has been validated by a number of valid experiments:
  • The Hafele-Keating atomic clock experiment involved flying atomic clocks at high speeds on aircraft and comparing them with synchronised clocks on the surface of Earth. This experiment helped validate time dilation.
  • Observations of significantly more muons on Earths surface compared to classical predictions due to the time dilation of the muon’s lifespans further validated special relativity.
  • Observations of momentum dilation of particles travelling at high velocities in particle accelerators.
Show Worked Solution
  • Theories in science must be consistent with observations and supported by independent, valid experiments in order to be validated.
  • Special relativity has been validated by a number of valid experiments:
  • The Hafele-Keating atomic clock experiment involved flying atomic clocks at high speeds on aircraft and comparing them with synchronised clocks on the surface of Earth. This experiment helped validate time dilation.
  • Observations of significantly more muons on Earths surface compared to classical predictions due to the time dilation of the muon’s lifespans further validated special relativity.
  • Observations of the momentum dilation of particles travelling at high velocities in particle accelerators.

♦ Mean mark 54%.

PHYSICS M7 2022 HSC 30

In a thought experiment, light travels from `X` to a mirror `Y` and back to `X` on a moving train carriage. The path of the light relative to an observer on the train is shown.
 


 

Relative to an observer outside the train, the path of the light is shown below, at three consecutive times as the train carriage moves along the track.
 

  1. Describe qualitatively how the constancy of the speed of light and the thought experiment above led Einstein to predict time dilation.  (3 marks)

  1. The train is travelling with a velocity  `v=0.96 c`. To the observer inside the train, the return journey for the light between `X` and `Y` takes 15 nanoseconds.

  2. How long would this return journey take according to the observer outside the train?  (3 marks)

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a.  Consider each observer:

  • The observer on the train sees light travel a distance from `X` to `Y` and back to `X` again.
  • The observer outside the train sees light travel a longer path due to the horizontal motion of the train.
  • As the speed of light is constant for both observers, the observer outside the train must observe the light pulse to take a longer time, `t=(text{distance})/(c).`
  • This provided the basis for Einstein’s predictions of time dilation.

b.   53.6  nanoseconds

Show Worked Solution

a.  Consider each observer:

  • The observer on the train sees light travel a distance from `X` to `Y` and back to `X` again.
  • The observer outside the train sees light travel a longer path due to the horizontal motion of the train.
  • As the speed of light is constant for both observers, the observer outside the train must observe the light pulse to take a longer time, `t=(text{distance})/(c).`
  • This provided the basis for Einstein’s predictions of time dilation.

♦ Mean mark (a) 48%.

b.    \(t=\dfrac{t_0}{\sqrt{1-\dfrac{v^2}{c^2}}}=\dfrac{15}{\sqrt{1-\dfrac{(0.96 c)^2}{c^2}}} = 53.6\ \text{nanoseconds} \)

PHYSICS, M7 2019 HSC 27

  1. Outline a thought experiment that relates to the prediction of time dilation.   (3 marks)
  1. Outline experimental evidence that validated the prediction of time dilation.   (3 marks)
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a.    Consider a train moving at a high speed:

  • There are two observers, one on the train and one stationary outside the train. 
  • A pulse of light starts from one side of the carriage and then reflects off a mirror on the other side of the carriage returning to its source. 
  •  The stationary observer, outside the train, will observe the light travel in a triangular path. This is longer than the path observed by the observer on the train.
  • The speed of light is constant for both observers.
  • the observer outside the train measures a longer, dilated time for the light pulse to travel. This demonstrates time dilation. 

b.    Experiment predicting time dilation:

  • Muons are particles in the upper atmosphere produced by cosmic rays that travel at a high speed greater than 0.99c and have a short half-life. 
  • The amount of muons striking the ground in a particular area at the top of a mountain was measured.
  • Using this data the number of muons expected to reach the ground at sea level was predicted (assuming no relativistic effects).
  • The actual number observed at sea level was greater than predicted.
  • This is consistent with an increase in the muons half life due to time dilation.

Answers could also reference:

  • The Hafele-Keating atomic clock experiment.
  • Evidence from particle accelerators.
Show Worked Solution

a.    Consider a train moving at a high speed:

  • There are two observers, one on the train and one stationary outside the train. 
  • A pulse of light starts from one side of the carriage and then reflects off a mirror on the other side of the carriage returning to its source. 
  • The stationary observer, outside the train, will observe the light travel in a triangular path. This is longer than the path observed by the observer on the train.
  • The speed of light is constant for both observers.
  • the observer outside the train measures a longer, dilated time for the light pulse to travel. This demonstrates time dilation. 

b.    Experiment predicting time dilation:

  • Muons are particles in the upper atmosphere produced by cosmic rays that travel at a high speed greater than 0.99c and have a short half-life. 
  • The amount of muons striking the ground in a particular area at the top of a mountain was measured.
  • Using this data the number of muons expected to reach the ground at sea level was predicted (assuming no relativistic effects).
  • The actual number observed at sea level was greater than predicted.
  • This is consistent with an increase in the muons half life due to time dilation.

Answers could also reference:

  • The Hafele-Keating atomic clock experiment.
  • Evidence from particle accelerators.

PHYSICS, M7 2021 HSC 28

A spaceship travels to a distant star at a constant speed, `v`. When it arrives, 15 years have passed on Earth but 9.4 years have passed for an astronaut on the spaceship.

  1. What is the distance to the star as measured by an observer on Earth?   (3 marks)
  1. Outline how special relativity imposes a limitation on the maximum velocity of the spaceship.   (2 marks)
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a.    12 ly

b.    According to special relativity, as ` v\ →\ c`:

  •   the momentum of the spaceship approaches infinity
  •   the force required to accelerate the spaceship approaches infinity
  •   maximum velocity is limited to the speed of light.
Show Worked Solution

a.    `t_v =t_0/sqrt((1-(v^2)/(c^2)))`

`15=9.4/sqrt((1-(v^2)/(c^2)))`

`1-(v^2)/(c^2)` `=((9.4)/(15))^2`  
`(v^2)/(c^2)` `=1-((9.4)/(15))^2`  
`v^2` `=0.60729c^2`  
`v` `=0.779c`  

  
Distance to star from Earth observer:

`s=ut=0.779 xx15=12\ text{ly}`
 


♦ Mean mark part (a) 51%.

b.  According to special relativity, as ` v\ →\ c`:

  • the momentum of the spaceship approaches infinity
  • the force required to accelerate the spaceship approaches infinity
  • maximum velocity is limited to the speed of light.