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Special Relativity: Length Contraction Problem

  1. Apr 25, 2017 #1
    Hello! I have a small problem with a task professor gave us. I tried many options (you will see below) but I cannot seem to get the right solution. Any advice or guideline how to solve this would be really helpful. In advance I thank you for helping me.

    1. The problem statement, all variables and given/known data
    Our professor of physics has sense of humor, so he represented this task: starship Enterprise captain measures his ship to be ##4500## meters long. Enterprise passes Earth with velocity of ## 0.8 c_0 ##. In opposite direction, starship Galactica flies by with velocity of ##0.9 c_0##. Compute how long will Enterprise appear to Galactica's captain.

    2. Relevant equations
    Pretty obvious, this task will implement length contraction formula $$ L = \sqrt{1 - \frac{v^2}{c_0^2}} L_0 $$. Also, according to solutions, final result should be ##684## meters.

    3. The attempt at a solution
    OK, so I began with computing Enterprise's velocity according to Galactica. Using ## v_e' = \frac{0.8c_0 - 0.9c_0}{1 - \frac{0.9c_0 \cdot 0.8c_0}{c_0^2}} = 0.35714c_0 ## I though I should just simply insert this into length contraction formula. It returned ##3608.1## meters.
    Now I decided to use slightly different procedure. I used ## L = \sqrt{1 - \frac{v_e' v_{galactica}}{c_0^2}} L_0 ## but it returned ##3706.9## meters.

    At this point I lost all hope. I really wish someone would be able to help me with this one.
     
  2. jcsd
  3. Apr 25, 2017 #2

    PeroK

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    The two ships must have a relative velocity of close to ##c##. How can ##0.357c## be right?
     
  4. Apr 26, 2017 #3
    I used equation ## v_e' = \frac{v_e - v_g}{1 - \frac{v_e v_g}{c_0^2}} ## to determine velocity of Enterprise according to Galactica.. Same equation appeared in previous example and in our textbook so I assumed it must be correct. It returned ## v_e' = 1.07068 \cdot 10^8 m/s ##. I think I must be missing an important step in all of this but I can't seem to find it.
     
  5. Apr 26, 2017 #4

    PeroK

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    Forget relativity for a moment. If two cars are coming at you from opposite directions at ##80km/h## and ##90km/h## respectively. The relative speed of the two cars is?

    a) ##170km/h##

    b) ##10km/h##
     
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