Derivation of Length Contraction (special relativity)

In summary, the derivation for length contraction involves the time dilation equation and the fact that the time taken by the light to do one back and forth is equal to the gamma*time. This ultimately leads to the conclusion that the moving length is equal to the normal length divided by the gamma factor.
  • #1
Chewy0087
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Homework Statement



Show the derivation for legnth contraction, given that from the example given;

The example shows a moving light clock - two walls which are moving in a direction at speed u and a beam of light from wall A to wall B then back to A and so forth.

From Ao (starting point) to B1 (distance moved after the beam has got there)

=t1 = "Moving" legnth / c - u

And from B1 - A2

t1 (2) = "Moving" Length / c + u

Homework Equations



t subscript 1 = 2L / c * sqroot (1 - u²/c²)

Basically the time dilation equation is given (and i managed to work out in an earlier question)

The Attempt at a Solution



Well I added up t1 and t1 (2), to get the time taken by the light to do one back and forth getting;

t1 + t1(2) = gamma*time = 2 * Moving Length / c*(1- u²/c²)

Making Moving length the subject;

Moving Length = (c*(gamma*time))*(1-u²/c²)

Put gamma into the equation;

Moving Length = (c/2)*(1-u²/c²)*2L/(c*sqrt(1-u²/c²))

And from here I'm totally stuck, I do know that the final answer is "Normal" length / gamma but really stuck taking the final steps... :/

Any help would be appreciated
 
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  • #2
Chewy0087 said:
t1 + t1(2) = gamma*time = 2 * Moving Length / c*(1- u²/c²)

Making Moving length the subject;

Moving Length = (c*(gamma*time))*(1-u²/c²)
You're missing a factor of 2 here, but you seem to have got it right on the next step.

Put gamma into the equation;

Moving Length = (c/2)*(1-u²/c²)*2L/(c*sqrt(1-u²/c²))

And from here I'm totally stuck, I do know that the final answer is "Normal" length / gamma but really stuck taking the final steps... :/

Any help would be appreciated
By "moving length", do you mean the wall separation measured in the lab frame or the moving (light clock) frame? Think about this.

As far as the math goes, you're nearly done. It's just a matter of canceling out terms from the numerator and denominator.
 

Related to Derivation of Length Contraction (special relativity)

1. What is the concept of length contraction in special relativity?

Length contraction is a phenomenon observed in special relativity where an object's length appears to decrease when it is moving at high speeds relative to an observer.

2. How is length contraction calculated in special relativity?

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

3. What does the derivation of length contraction involve?

The derivation of length contraction involves applying the principles of special relativity, including the constancy of the speed of light and the relativity of simultaneity, to derive the formula for length contraction.

4. Why is length contraction important in special relativity?

Length contraction is important in special relativity because it helps explain why objects moving at high speeds appear to be shorter than they actually are. It also plays a crucial role in understanding the effects of time dilation and the relativity of simultaneity.

5. Is length contraction a real physical phenomenon or just a mathematical concept?

Length contraction is a real physical phenomenon that has been experimentally verified. It is not just a mathematical concept, but rather a consequence of the principles of special relativity and the nature of spacetime.

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