# Length Contraction Relativity Question

• goldilocks
In summary, the conversation is about a question regarding length contraction in relativity. The person is unsure if they have solved the problem correctly and wonders if they should have used Taylor's expansion. They have attempted a solution using the formula for length contraction but have not taken into account the small speed compared to the speed of light. The expert suggests looking up the Taylor expansion for a more accurate calculation.
goldilocks
Hello, I am not sure if I am doing this right or not. It seems to simple to be the right aswer, I feel I must have gone wrong somewhere. Many thanks :-) xxx

## Homework Statement

Length Contraction Relativity Question?

An aeroplane is of length L’(m) precisely when measured in its rest frame at the airport. In what percentage it will contract if measured by the same ground-based observer if it was flying past at 900 km.p.h?
[Here you may need Taylor’s expansion]

## Homework Equations

Lmoving= Lrest x SQR[1 - (V/C)^2]

## The Attempt at a Solution

I have tried to start this. I used Lmoving= Lrest x SQR[1 - (V/C)^2]

Lrest = L'(m)

Lmoving = L'(m)
= L'(m) x SQR(1-(250^2/(3x10^8)^2)
= L'(m) x SQR (1- 6.9445x10^-13)
= L'(m) x SQR (1)
= L'(m)

So there is no contraction as v is not near enough to c for realtivity to take effect. Is this right? I didn't use taylor expansion, so have I gone wrong somewhere. Thanks a lot. xxx

goldilocks said:
So there is no contraction as v is not near enough to c for realtivity to take effect. Is this right?
No. Since the speed is small compared to c, the effect will be small. But you can still calculate it.

I didn't use taylor expansion, so have I gone wrong somewhere.
Hint: Look up the Taylor expansion of $\sqrt{1 + x}$, where x << 1.

## 1. What is length contraction in relativity?

Length contraction is a phenomenon in which an object's length appears shortened when it is moving at high speeds relative to an observer. This is a consequence of Einstein's theory of relativity, which states that the laws of physics are the same for all observers in uniform motion.

## 2. How does length contraction occur?

Length contraction occurs due to the relativity of simultaneity, where two observers in relative motion will not agree on the timing of events. This leads to a difference in the measurements of an object's length between the two observers.

## 3. Can length contraction be observed in everyday life?

No, length contraction can only be observed at extremely high speeds, such as those reached by particles in particle accelerators. At everyday speeds, the effects of length contraction are too small to be noticeable.

## 4. What is the formula for calculating length contraction?

The formula for calculating length contraction is L = L0 * √(1 - v2/c2), where L is the measured length, L0 is the rest length, v is the velocity of the object, and c is the speed of light.

## 5. How does length contraction relate to time dilation?

Length contraction and time dilation are two sides of the same coin in relativity. As an object's length appears shortened due to its high speed, time also appears to slow down for the object. This is known as time dilation, and it is another consequence of Einstein's theory of relativity.

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