University of Wisconsin Green Bay

An astronaut on a spaceship traveling at a constant velocity away from Earth travels for 12 years before she encounters another solar system. According to her colleagues on Earth, the trip to the new solar system took 17 years. How fast did the astronaut travel? How far from Earth is the new solar system?

  • In this problem, an observer on the spaceship measures how long it takes her to make a trip, and an observer on the Earth times the same trip. Any problem that relates the measurements of two observers measuring the same thing is a relativity problem.

    In the second question, you are asked to relate distance to the closely related quantities of speed and time. The second question is just a definition.

  • Step 1

    Observers on Earth and Spaceship

    time of travel direction of travel direction of travel arrow arrow 17 y 12 y reference frames reference frames direction of motion direction of motion
  • Spaceship and time equation

    In this case, both observers measure the time it takes the astronaut and spaceship to travel, and so the problem is understood through the relativistic time equation regardless of what quantity is requested in the solution.

    how fast
  • 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.


    Step 2

    Now that you know how fast the spaceship is traveling, it is only a definition problem to determine how far it goes.

    Two cautions:

    - Because you want distance in conventional units such as meters, make sure to use velocity in units of m/s and not in units of c.
    - With the exception of the three relativistic equations, all physics equations are worked within a single reference frame. Make sure to use distance and time measurements within the same frame in the velocity equation.

    The definition of velocity is

    The new solar system is 1.1 x 1017 m from Earth. No further mathematical solution is needed in this problem.

  • division 0.50 sign square root 0.71 c 17 y time in s final answer 2.62
  • Regardless of what you were asked to find, this is a relativistic time problem because two observers in different reference frames both time the same event. In this case, it is the astronaut on the spaceship that does the action being timed, and so the spaceship is the proper frame (and measures the shorter time.)

    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.

    In the second question, you were asked to related length to time. This is not a relativistic question because it does not involve two observers measuring the same thing. Rather a single observer measured two different things. All physics equations that related different quantities must be worked in a single reference frame.