so I'm working on length contraction in relativity theory. I feel pretty confident time dilation and have not really gone over Lorentz Transformations that much.
The question itself lies at the bottom if you want to skip to there. The following just shows my process and conclusion
So, I'm working on a thought experiment where a snake is moving relative to Observer 2. I'm trying to determine the relation between the lengths of each reference frame: snake and Observer 2. Here is a (very messy) illustration:
Okay so the variables for each reference frame would be: $$ S_1(snake): \Delta t_1 , L_1 $$ $$ S_2: \Delta t_2 , L_2 $$
And
Event 1 = the measure at the point at the rear side of the snake. Event 2 = the measure at the point at the front side of the snake.
I am just going to assume that the following equation is true for now after having previously proved it in a time dilation experiment:
$$ \Delta t = \gamma \Delta t_o $$
Since in both reference frames, $S_1 , S_2$, Event 1 and Event 2 do not occur at the same point, we can say the proper time $\Delta t_o$ cannot be found in either of the frames. Likewise, it will be found in some other frame. Therefore, from the equation found in the time dilation experiment, we can reasonably say: $$\Delta t_1 = \gamma \Delta t_o$$ $$\Delta t_2 = \gamma \Delta t_o $$ Where the values of $\gamma$ in each equation are equal because V would be equal in each equation. Then we conclude: $$\Delta t_1 = \Delta t_2 $$ From there, since $L = V \Delta t$, we can say:
$$L_2 = V \Delta t_2 $$
Since both times values are equivalent: $$L_2 = V \Delta t_1 $$ Then, we can replace $\Delta t_1$ with $L_1 \over V$. Plugging that back into the equation: $$L_2 = V {L_1 \over V} $$ and $$L_2 = L_1 $$ This seems to be rather uneventful, and I'm assuming I did something wrong. It would make sense that if time can be different in different frames then length would be too. I'm pretty sure the reason this lame result occurred was because I assumed that the $\Delta t$ in each frame of reference was the same. The text I'm reading states that the following equation: $$L_2 = {L_1 \over \gamma}$$ I know I'm forgetting something, but I can't figure out what it is. Can anyone see what I'm doing wrong?
Thanks!