smc-3699-20-Length Contraction

PHYSICS, M7 2019 VCE 13 MC

Joanna is an observer in Spaceship \(\text{A}\), watching Spaceship \(\text{B}\) fly past at a relative speed of 0.943\(c\). She measures the length of Spaceship \(\text{B}\) from her frame of reference to be 150 m.

Which one of the following is closest to the proper length of Spaceship \(\text{B}\)?

50 m
150 m
450 m
900 m

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\(C\)

Show Worked Solution

\(l\)	\(=l_0\sqrt{1-\frac{v^2}{c^2}}\)
\(l_0\)	\(=\dfrac{l}{\sqrt{1-\frac{v^2}{c^2}}}\)
	\(=\dfrac{150}{\sqrt{1-0.943^2}}\)
	\(=450\ \text{m}\)

\(\Rightarrow C\)

PHYSICS, M7 2021 VCE 10

A new spaceship that can travel at 0.7\(c\) has been constructed on Earth. A technician is observing the spaceship travelling past in space at 0.7\(c\), as shown in Figure 10. The technician notices that the length of the spaceship does not match the measurement taken when the spaceship was stationary in a laboratory, but its width matches the measurement taken in the laboratory.

Explain, in terms of special relativity, why the technician notices there is a different measurement for the length of the spaceship, but not for the width of the spaceship. (2 marks)

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If the technician measures the spaceship to be 135 m long while travelling at a constant 0.7\(c\), what was the length of the spaceship when it was stationary on Earth? Show your working. (2 marks)

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a. By Einstein’s theory of special relativity when travelling at high speeds, length contraction occurs.

Length contraction only occurs in the direction of motion thus the technician measures a different measurement for the length but not the width of the spaceship.

b. \(l_o=189\ \text{m}\)

Show Worked Solution

a. By Einstein’s theory of special relativity:

→ When travelling at high speeds, length contraction occurs.

→ Length contraction only occurs in the direction of motion thus the technician measures a different measurement for the length but not the width of the spaceship.

b.	\(l\)	\(=l_0\sqrt{1-\frac{v^2}{c^2}}\)
	\(l_0\)	\(=\dfrac{l}{\sqrt{1-\frac{v^2}{c^2}}}\)
		\(=\dfrac{135}{\sqrt{1-\frac{(0.7c)^2}{c^2}}}\)
		\(=\dfrac{135}{\sqrt{1-0.7^2}}\)
		\(=189\ \text{m}\)

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 VCE 10

A proton in an accelerator beamline of proper length 4.80 km has a Lorentz factor, \(\gamma\), of 2.00.

Calculate the speed of the proton relative to the beamline in terms of \(c\), the speed of light in a vacuum. Give your answer to three significant figures. (3 marks)

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Calculate the length of the beamline in the reference frame of the proton. (1 mark)

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a. \(0.866c\)

b. \(2.4\ \text{km}\)

Show Worked Solution

a. \(\dfrac{1}{\sqrt{1-\frac{v^2}{c^2}}}\)	\(=2\)
\(\sqrt{1-\dfrac{v^2}{c^2}}\)	\(=\dfrac{1}{2}\)
\(1-\dfrac{v^2}{c^2}\)	\(=\dfrac{1}{4}\)
\(\dfrac{v^2}{c^2}\)	\(=\dfrac{3}{4}\)
\(v^2\)	\(=\dfrac{3}{4}c^2\)
\(v\)	\(=0.866c\)

b.	\(l\)	\(=l_0\sqrt{1-\dfrac{v^2}{c^2}}\)
		\(=4.8 \times \sqrt{1-\dfrac{(0.886c)^2}{c^2}}\)
		\(=4.8 \times 0.5\)
		\(=2.4\ \text{km}\)

PHYSICS, M7 2023 HSC 19 MC

The diagram represents the distribution of positive charges in identical wires when no current is flowing.

Equal currents then flow in each wire, but in opposite directions. These currents are considered conventionally as the flow of positive charge.

Which diagram represents the charge distribution in the wires, from the frame of reference of a positive charge in wire \(Y\) ?

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\(B\)

Show Worked Solution

→ The currents flowing through Wire \(Y\) are all travelling in the same direction and at the same speed, thus they are all within the same inertial frame of reference and will not experience any relativistic effects when viewed from the frame of reference of a positive charge in \(Y\).

→ Therefore, the separation of the charges in \(Y\) from the frame of reference of \(Y\) will not change.

→ As the positive charges in \(X\) are moving in the opposite direction to the positive charges in \(Y\), from the frame of reference of a positive charge in \(Y\), length contraction will be observed in \(X\).

→ Therefore, the separation of the positive charges in \(X\) from the frame of reference of \(Y\) will be shorter so the charges appear closer together.

\(\Rightarrow B\)

♦ Mean mark 44%.

PHYSICS, M7 EQ-Bank 3 MC

A spaceship sitting on its launch pad is measured to have a length `L`. This spaceship passes an outer planet at a speed of 0.95`c`.

Which observations of the length of the spaceship are correct?

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`A`

Show Worked Solution

→ An observer in the frame of reference of the spaceship will measure the proper length of the spaceship.

→ An observer on the planet will observe relativistic length contraction and will observe the length to be shorter than `L`.

`=>A`

PHYSICS, M7 2015 HSC 16 MC

Astronauts travel at a velocity of 0.9 `c` to Alpha Centauri. Newtonian physics predicts that this journey would take 4.86 years.

How many years will the journey take in the frame of reference of the astronauts?

0.923
1.54
2.12
11.1

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`C`

Show Worked Solution

The Newtonian distance to Alpha Centauri: `4.86 xx0.9 = 4.373` light years.

Finding the relativistic distance due to length contraction:

`l`	`=l_(0)sqrt(1-(v^2)/c^2)`
	`=4.373xx sqrt(1-((0.9c)^2)/c^2)`
	`=4.373 xxsqrt(1-0.9^2)`
	`=1.906\ text{ly}`

This will take `(1.906)/(0.9) = 2.12` years.

`=>C`

Mean mark 59%.

PHYSICS, M7 2017 HSC 20 MC

The length of a spaceship is measured by an observer to be 3.57 m as the spaceship passes with a velocity of 0.7`c`.

At what velocity would the spaceship be moving relative to the observer if its measured length was 2.5 m?

0.490`c`
0.707`c`
0.714`c`
0.866`c`

Show Answers Only

`D`

Show Worked Solution

`l`	`=l_(0)sqrt((1-(v^(2))/(c^(2))))`
`3.57`	`=l_(0)sqrt((1-((0.7c)^(2))/(c^(2))))`
`3.57`	`=l_(0)sqrt(1-0.7^2)`
`l_(0)`	`=(3.57)/(sqrt(1-0.7^2))`
	`=5.0 text{m}`

If the measured length were to be 2.5 m:

`2.5`	`=(5.0)sqrt((1-(v^(2))/(c^(2))))`
`(1)/(2)`	`=sqrt(1-(v^(2))/(c^(2)))`
`1-(v^(2))/(c^(2))`	`=(1)/(4)`
`(v^(2))/(c^(2))`	`=(3)/(4)`
`v^2`	`=(3)/(4)c^2`
`v`	`=0.866c`

`=>D`

PHYSICS, M7 2018 HSC 16 MC

When a train is at rest in a tunnel, the train is slightly longer than the tunnel.

In a thought experiment, the train is travelling from left to right fast enough relative to the tunnel that its length contracts and it fits inside the tunnel.

An observer on the ground sets up two cameras, at `X` and `Y`, to take photos at exactly the same time. The photos show that both ends of the train are inside the tunnel.

A passenger travelling on the train at its centre can see both ends of the tunnel and is later shown the photos.

From the point of view of the passenger, what is observed and what can be deduced about the photos?

The tunnel's length contracts so the train does not fit, and photo 2 is taken before photo 1.
The tunnel's length contracts so the train does not fit, and photos 1 and 2 are taken at the same time.
The tunnel appears to expand due to length contraction of the train, allowing it to fit in the tunnel, and photo 1 is taken before photo 2 .
The tunnel appears to expand due to length contraction of the train, allowing it to fit in the tunnel, and photos 1 and 2 are taken at the same time.

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`A`

Show Worked Solution

→ From the frame of reference of an observer on the train, the tunnel’s length will contract (all objects outside the train will appear length contracted).

→ Due to the relativity of simultaneity, the photo that the observer on the train sees as taken first is the photo taken at the end of the tunnel that the train was moving towards at that moment.

`=>A`

♦♦♦ Mean mark 36%.

PHYSICS, M7 2018 HSC 9 MC

A hypothetical journey to a distant star might be accomplished in an astronaut's life span by travelling at relativistic speeds.

What is the key concept which underpins such a hypothetical journey?

From the frame of reference of the spacecraft making the journey, the distance to the star is less.
Clocks on the spacecraft run slower and hence the rate at which fuel is used for the journey is decreased.
Relativistic effects on the spacecraft reduce its mass, making it possible to accelerate to the speeds needed for such a journey.
The relativistic increase in the mass of the fuel on board makes it possible to complete a longer journey with less fuel.

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`A`

Show Worked Solution

→ Due to the relativistic effect of length contraction, the distance required to travel will decrease.

→ This reduces the travel time.

`=>A`

PHYSICS M7 2022 HSC 20 MC

In a thought experiment, a car is travelling at a uniform velocity of `0.4 c`. The diagram shows one of the car's wheels as it rolls past a stationary observer at `X`.

Consider the instantaneous velocity of different points on the car's wheel relative to the ground. Assume that there is no slippage of the tyre on the road.

At the instant the centre of the wheel, `Q`, passes `X`, how would the observer describe the relativistic length contraction at points `P, Q` and `R` ?

It is the same at `P, Q` and `R`.
It is zero at `P` and greatest at `R`.
It is equal at `P` and `R`, and least at `Q`.
It is zero at `P` and the same value at `Q` and `R`.

Show Answers Only

`B`

Show Worked Solution

Consider Point `P`:

→ Point `P` is moving at 0.4`c` left relative to point `Q`.

→ Point `P` has zero instantaneous velocity relative to the ground.

→ Zero relativistic length contraction will be observed at `P`.

Consider Point `R`:

→ Point `R` is moving at 0.4`c` right relative to point `Q`.

→ Point `R` has an instantaneous velocity of 0.8`c` relative to the ground.

→ Maximum relativistic length contraction will be observed at `R`.

`=>B`

♦♦♦ Mean mark 30%.

PHYSICS, M7 2020 HSC 17 MC

In a thought experiment, observer `X` is on a train travelling at a constant velocity of 0.95c relative to the ground. Observer `Y` is standing on the ground outside the train. As observer `X` passes observer `Y`, observer `X` sends a short light pulse towards the sensor.

Which statement about the light pulse is correct as observed by `X` or `Y` in their respective frames of reference?

Its velocity observed by `Y` is 0.05c.
`X` sees it travel a shorter distance to the sensor than `Y`.
`X` sees it take a longer time to reach the sensor than `Y`.
Both `X` and `Y` see it travel the same distance in the same amount of time.

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`C`

Show Worked Solution

`Y` observes the length of the carriage to be shorter than `X` does due to length contraction.

As the speed of light is equal for both observers, `X` sees the light pulse take a longer time to reach the sensor.

`=>C`