Key 4

Up a level : Lorentz transformation of space
  1. The asteroid question.
    1. Calculate the gamma factor.
      \displaystyle \gamma =\frac{1}{{\sqrt{{1-{{{\left( {\frac{v}{t}} \right)}}^{2}}}}}}=\frac{1}{{\sqrt{{1-{{{0.8}}^{2}}}}}}=\frac{5}{3}=1.67
    2. How long is the spacecraft as seen from the asteroid?
      Here we can simply use the length contraction formula:
      \displaystyle L=\frac{{{{L}_{0}}}}{\gamma }=\frac{{100}}{{({5}/{3}\;)}}=60\text{ m}
    3. How long is the asteroid as seen from the spacecraft?
      Same here. We get:
      \displaystyle L=\frac{{{{L}_{0}}}}{\gamma }=\frac{{900}}{{({5}/{3}\;)}}=540\text{ m}
    4. Calculate the distance between the light flashes as seen from the spacecraft.
      Now we are dealing with two events in spacetime. Now we need to use the Lorentz transformation. We get
      \displaystyle {x}'=\gamma (x-vt)=\frac{5}{3}(900-0.8c\cdot 0)=1500\text{ m}
      The proper time is t=0, because the two events happen at the same time in the frame of the asteroid.
Up a level : Lorentz transformation of spaceLast modified: Jun 6, 2025 @ 16:46