Grade 12

Advanced

7 min

Lesson 3 of 12

Length Contraction and Relativistic Space

Not Started

Investigating how the length of an object appears to shorten in the direction of motion as it approaches the speed of light.

If you were traveling at 99% the speed of light, a 100-meter long spaceship would appear to be only 14 meters long to a stationary observer. Is the ship actually shrinking, or is space itself playing tricks on our perception?

Learning Objectives

1.Explain length contraction as a necessary consequence of the constancy of the speed of light
2.Distinguish between proper length ( $L_0$ ) and contracted length ( $L$ )
3.Apply the length contraction formula to solve high-speed interstellar travel problems

The Relativity of Space

In classical physics, we assume space is a fixed stage. However, Einstein’s Special Relativity reveals that space is flexible. Because the speed of light ( $c$ ) must be the same for all observers, something has to give when objects move at high speeds. If time slows down for a moving object (time dilation), then the distance it travels must also change to keep the math consistent. This phenomenon is called Length Contraction. It states that the length of an object is measured to be shorter when it is moving relative to an observer. Crucially, this 'shrinking' only occurs in the direction of motion. If a rocket flies horizontally, it gets thinner from nose to tail, but its height remains exactly the same.

Quick Check

If a rectangular box flies past you at 90% the speed of light moving left-to-right, which dimension(s) will appear shorter?

Contraction only occurs in the direction of travel.

Answer

Only the width (the dimension parallel to the left-to-right motion).

Proper Length vs. Contracted Length

To solve relativity problems, you must identify the two types of length. Proper Length (

L_0

) is the length of an object measured by an observer who is at rest relative to that object. For example, if you are on a bus, you measure its proper length. Contracted Length (

L

) is the length measured by an observer who sees the object moving past them. The relationship is defined by the formula:

\begin{gathered}L = L_0 \\sqrt{1 - \\frac{v^2}{c^2}}\end{gathered}

Because the term inside the square root is always less than 1 (for any speed

v < c

), the measured length

L

will always be smaller than the proper length

L_0

. This isn't an optical illusion; for the stationary observer, the object occupies less physical space.

A meter stick ( $L_0 = 1.0$ m) flies past you at $v = 0.8c$ . How long does it appear to be?

1. Identify the variables: $L_0 = 1.0$ , $v = 0.8c$ . 2. Set up the formula: $L = 1.0 \\sqrt{1 - (0.8)^2}$ . 3. Calculate the square: $0.8^2 = 0.64$ . 4. Subtract from one: $1 - 0.64 = 0.36$ . 5. Take the square root: $\\sqrt{0.36} = 0.6$ . 6. Final result: $L = 1.0 \\times 0.6 = 0.6$ meters.

The Interstellar Perspective

Length contraction applies not just to objects, but to the distance between points in space. Imagine a star system 10 light-years away. To an observer on Earth, that distance is a 'proper length' because the stars aren't moving relative to Earth. However, to an astronaut traveling at $0.95c$ , that 10 light-year gap is moving toward them at high speed. Consequently, the astronaut measures the distance to be much shorter. This explains why high-speed travel is possible: the traveler doesn't just get there faster because they are moving quickly; they get there faster because the distance they have to travel actually shrinks in their frame of reference.

A ship travels to a star 4.0 light-years away (as measured by Earth) at a speed of $v = 0.9c$ . What distance does the pilot measure?

1. Earth measures the proper length $L_0 = 4.0$ ly. 2. The pilot is in motion relative to the distance, so they measure $L$ . 3. $L = 4.0 \\sqrt{1 - (0.9)^2} = 4.0 \\sqrt{1 - 0.81}$ . 4. $L = 4.0 \\sqrt{0.19} \\approx 4.0 \\times 0.436$ . 5. $L \\approx 1.74$ light-years. The pilot sees the journey as less than half the distance Earth sees.

Quick Check

If an astronaut is measuring the length of their own ship while flying at 0.99c, will they measure $L$ or $L_0$ ?

Who is moving relative to the object being measured?

Answer

They will measure $L_0$ (Proper Length) because they are at rest relative to the ship.

Muons are subatomic particles created in the upper atmosphere, 10 km above Earth. They travel at $0.999c$ . Their lifespan is so short they should decay before hitting the ground. However, they reach the surface.

1. From the Muon's frame, the 10 km atmosphere is rushing toward it. 2. Calculate the contracted height of the atmosphere: $L = 10 \\sqrt{1 - (0.999)^2}$ . 3. $L = 10 \\sqrt{1 - 0.998001} = 10 \\sqrt{0.001999} \\approx 0.447$ km. 4. Because the atmosphere 'shrinks' to only 447 meters for the muon, it can reach the ground before its short life ends.

Key Takeaways

1Length contraction is the shortening of an object's length in the direction of its motion relative to an observer.
2The Proper Length ( $L_0$ ) is always the longest measurement of an object, taken by an observer at rest relative to it.
3The formula is $L = L_0 \\sqrt{1 - \\frac{v^2}{c^2}}$ , where $v$ is the relative velocity and $c$ is the speed of light.
4Length contraction and time dilation are two sides of the same coin, ensuring the speed of light remains constant for everyone.

Quick Check

Multiple Choice

In which direction does an object appear to contract?

Multiple Choice

As the velocity of an object approaches the speed of light ( $c$ ), what happens to its measured length $L$ according to a stationary observer?

True/False

An astronaut on a high-speed rocket will see their own reflection in a mirror as being thinner than usual.

What's Next?

Review Tomorrow

In 24 hours, try to explain to a friend why a muon (a tiny particle) 'thinks' the Earth's atmosphere is much thinner than we measure it to be.

Practice Activity

Calculate the contracted length of a 100m spaceship traveling at $0.99c$ and compare it to the hook's claim of 14 meters.

Browse

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Length Contraction and Relativistic Space

Learning Objectives

The Relativity of Space

Proper Length vs. Contracted Length

The Interstellar Perspective

Key Takeaways

Quick Check

What's Next?

Length Contraction and Relativistic Space

Learning Objectives

The Relativity of Space

Proper Length vs. Contracted Length

The Interstellar Perspective

Key Takeaways

Quick Check

What's Next?