University of Wisconsin Green Bay

An observer on Earth watches two spaceships. According to his measurements, spaceship A is traveling away from Earth at half the speed of light, and spaceship B is approaching Earth at 0.75 c. How fast is spaceship A moving according to the crew on spaceship B?

  • In this problem, an observer on Earth and an observer on spaceship B both measure the speed of spaceship A. Any problem that relates the measurements of two observers measuring the same thing is a relativity problem.

  • observer on Earth spaceship A spaceship A crew on spaceship B Earth observer arrow speed of A speed of B observer B
  • Spaceship and time equation

    In this case, both observers measure the speed of spaceship A. Therefore, this is a relativistic velocity addition problem regardless of what quantity is requested in the solution.

    how fast u u prime v
  • Step 1



    The first step in solving a velocity addition problem is to identify which is the prime observer and which is the unprime observer. There is no proper frame, so the choice is yours! However, the math is always easier if u (rather than u’) is the unknown quantity. Once the prime and unprime frame are identified, symbols can be assigned to the given quantities.

    prime frame unprime frame 0.5 c v


    Step 2

    According to an observer on spaceship B, spaceship A approaches with a speed of 0.91 c. No further mathematical solution is necessary.

    division u1.375 0.91 c
  • Regardless of what you were asked to find, this is a relativistic velocity problem because two observers in different reference frames both measure the speed of the same thing. There is no proper frame in this situation, so the prime and unprime frames for the observers need to be chosen and clearly identified.

    The numerical answer makes sense. First, we know that velocity will never be greater than c. Second, because spaceships A and B are approaching each other, we expect that the speed of A relative to B will be greater than the speed of either of the spaceships as measured by an observer on Earth.