University of Wisconsin Green Bay

A spaceship is 75 m long according to an astronaut on the ship. As it travels away from Earth, scientists on the ground measure the length of the ship to be 51 m. How fast is the spaceship traveling away from Earth?

  • In this problem, an observer on the spaceship measures the length of the ship, and an observer on the Earth also measures the length of the ship. Any problem that relates the measurements of two observers measuring the same thing is a relativity problem.

  • Step 1

    Observers on Earth and Spaceship

    spaceship ship lengths direction of travel arrow arrow lengths lengths reference frames reference frames direction of motion direction of motion
  • Spaceship and length equation

    In this case, both observers measure the length of the spaceship, and so the problem is understood through the relativistic length equation regardless of what quantity is requested in the solution.

    mg sin(15.5^0)
  • The spaceship is traveling at 73% the speed of light (or 2.2 x 108 m/s) away from Earth. No further mathematical solution is needed in this problem.

  • division 0.54 sign square root answer
  • Regardless of what you were asked to find, this is a relativistic length problem because two observers in different reference frames both measure the length of the same thing. In this case, it is the spaceship whose length is measured, and so the spaceship is the proper frame (and has the longer measurement.)

    Because the two measurements are noticeably different from each other, you expect that the relative speed of the frames should be a substantial fraction of the speed of light as found.