Have you ever been on a road with friend and argued about how far you have left to go? It's a common situation that can be easily resolved by checking the distance ahead, right? Well, that's true for everyday speeds. But what happens when you're travelling at super-fast speeds like close to the speed of light? This is where things get interesting! At these speeds, something called "length contraction" occurs, which causes two people in different reference frames to disagree on how far away something is. So, your friend might say you have 30km left while you think you only have 20km left!

What is length contraction?

Have you ever heard of length contraction? It's a weird phenomenon that happens at super-fast speeds. Basically, if an object is travelling really quickly with respect to a frame of reference, it can appear shorter than it actually is. The actual length of the object is called its "proper length" and is measured by an observer who is at rest relative to both points.

Here's another crazy thing: even though clocks measure different elapsed periods for the same procedure, the relative speed (distance divided by elapsed time) stays the same. But, the distance is affected by the relative motion of the observer. So, if two observers see different times, they must also measure different distances for the speed to remain constant.

This is where length contraction comes in. At near-light speeds, distances measured by different observers aren't the same. So, even if two people are measuring the same object, they might get different results!

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How do you calculate length contraction?

Having discussed what we mean by length contraction and proper length, let’s look at an example to explore how to calculate length contraction.

Let’s say a spaceship is moving at a velocity v that is close to the speed of light. An observer A on the earth and an observer B in the spaceship will observe different lengths for the distance covered by the spaceship.

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We know that the velocity of the spaceship is the same for all observers. If we calculate the velocity v relative to the earth-bound observer A, we get:

v = ΔL/Δt

Here, L0 is the proper length observed by the-bound observer A, while Δt is the time relative to the earth-bound observer A.

The velocity relative to the moving observer B is:

v = ΔL/Δt0

Here, Δt0 is the proper time observed by the moving observer B, while L is the distance observed by the moving observer B.

The two velocities are the same:

v = ΔL/Δt = ΔL/Δt0

We know from time dilation that t = t0. Entering this into the previous equation, we get:

v = ΔL/Δt = ΔL/Δt0 = ΔL/Δt

We also know that:

L = L0√1 − v2 c2

Inserting y, we get the equation for length contraction as shown below:

L = L0√1 − v2 c2

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Length contraction of an object’s length

One of the consequences of length contraction is that if an object is moving at a velocity near the speed of light, its length may be observed to be less than its proper length by an observer who is at rest relative to the motion. Let’s consider the following example.

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That's right. Let's say you have a 10cm stick that is at rest with respect to you, the observer. Then, when the stick starts moving at a very high speed close to the speed of light, its length will appear to be shorter than its proper length. This phenomenon is called length contraction.

The proper length is the length of an object when it is at rest relative to the observer. But when the object starts moving at a high speed, its length appears to be shorter. This is because the faster an object moves, the more its length appears to contract in the direction of its motion, as observed by an observer who is at rest relative to the object.

In fact, according to the theory of relativity, if the stick were to move at the speed of light, its length would theoretically become zero! This strange conclusion is a result of the fact that as an object moves closer to the speed of light, its mass increases, and it requires more and more energy to continue accelerating it. This means that it would be impossible to actually accelerate an object to the speed of light, but the theory of relativity still predicts that length contraction would occur as the object approaches that speed.

What is an example of length contraction?

A great example of length contraction is when an object is travelling through space, as in the following example.

Let’s imagine an observer is travelling from the blue planet to the red one and travelling at the speed of y=30.00. The distance between the two planets is 4,000 light-years as measured by an earth-bound observer. What is the distance relative to the observer on the spaceship in measured kilometres?

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Length contraction is a key concept in the theory of relativity, and it occurs when an object is moving at a high velocity relative to an observer. The proper length (L0) is the length of an object that is at rest relative to the observer, and it is the length that would be measured by an observer who is also at rest relative to the two endpoints of the object.

However, when the object starts moving at a high velocity, its length appears to be shorter than its proper length, as observed by an observer who is at rest relative to the object. The amount of length contraction depends on the velocity of the object and is given by the Lorentz factor y. The relationship between the proper length L0 and the observed length L is given by L = L0/γ.

It's important to note that length contraction is not just an optical illusion, but it is a real physical effect that is predicted by the theory of relativity. It has been observed in experiments involving high-energy particles, and it has also been used in the design of particle accelerators, where the contraction of the length of a particle beam is necessary to maintain its stability at high energies.

Length Contraction

What is length contraction?

Length contraction is the phenomenon that occurs when the length of an item traveling at a certain speed is measured to be shorter than its proper length.

Why does length contraction occur?

Length contraction is caused by the fact that the speed of light in a vacuum is constant in any frame of reference.

What are length contraction and time dilation?

When the length of an item travelling at a certain speed is measured to be less than its proper length, this is known as length contraction. Time dilation is the phenomenon by which time is measured differently for objects travelling through space than for stationary objects.

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