Length Contraction: Calculate Observed Length w/o and w/ Theory of Relativity

[Icheb]

A stick of 1m in length travels at v = 1/2 c along its axis away from the observer.

Question 1:
Show that the observer perceives the length of the stick to be shorter without theory of relativity. Calculate the length as perceived by him if he calculates it by the difference in length between both ends which have been photographed at the same time.

Question 2:
Use theory of relativity to solve the first question.

Regarding 1:
The light emitted by the far end of the stick has to be emitted a little bit sooner so that it reaches the observer at the same time as light from the front end of the stick. Since light travels at c towards the observer and the stick travels at 1/2 c away, it's necessary to know how fast 1/2 c (the difference between the speed of light and the speed of the stick) needs to cover 1m. The time it takes for this determines how much later the front end has to emit a photon so that it reaches the observer at the same time as a photon emitted by the far end.

Is that correct so far?

Now I'm not sure how to calculate the perceived length. I guess I have to find out where the photon of the far end has to be emitted relative to the front end, but how would I do that?

Regarding 2:
I'm not sure about the approach. I know how to use Lorentz transformation to get the new length, but would I include the effect calculated for question 1 as well or not?

 

[EngageEngage]

for question 2, you can use lorentz position transformations to find this but it is more difficult than to use the length contraction relation (if you have studied it). Ultimately, though, they will reduce to the same thing, and you will find:
$$l = l_{o}\sqrt{1-\frac{v^{2}}{c^{2}}}$$
If you use the lorentz transofmrations, be sure to remember that some relations must cancel ($$\Delta ? = 0$$ -- i leave the question mark for you to fill in) based on the way you must make measurements.

 

[Moetasim]

For question 1, you are correct in understanding that the light emitted from the far end of the stick will have to travel a longer distance to reach the observer at the same time as the light emitted from the front end. This is due to the relative motion between the stick and the observer. To calculate the perceived length, we can use the equation for length contraction: L' = L√(1-v^2/c^2), where L is the original length and v is the velocity of the stick. In this case, v = 1/2 c, so the perceived length will be L' = 1m√(1-(1/2)^2) = 0.866m. This means that the observer will perceive the stick to be shorter than its actual length of 1m.

For question 2, you can use the Lorentz transformation to calculate the observed length. The equation is L' = γ(L-vt), where γ is the Lorentz factor, L is the original length, v is the velocity of the stick, and t is the time interval between the front and far end of the stick emitting light. In this case, t = 0 since the light is emitted at the same time, so the equation simplifies to L' = γL. To calculate γ, we can use the equation γ = 1/√(1-v^2/c^2). Plugging in v = 1/2 c, we get γ = 1/√(1-(1/2)^2) = 1/√(3/4) = 2/√3. Therefore, the observed length using the theory of relativity is L' = (2/√3)1m = 0.577m. This is slightly different from the result in question 1 due to the inclusion of the time interval in the Lorentz transformation equation.