# University of Wisconsin Green Bay

An astronaut on a spaceship approaching Earth observes that it takes 2.0 hours for an asteroid to pass out of sight. Mission control on Earth measures the transit time of the asteroiod to be 25% longer than that. A second spaceship approaches Earth from the other side at one third the speed of the first spaceship. What is the transit time of the asteroid as measured on the second ship?

• In this problem, two observers on different spaceships (plus an observer on Earth) measure the time it takes an asteroid to cover a certain path. Any problem that relates the measurements of two observers measuring the same thing is a relativity problem.

You may feel uncomfortable by the lack of numbers in the problem. It is certainly true, for example, that you will need to know the velocities of the spaceships in order to calculate the time interval. In other words, this is a multi-part problem. Don't worry if you don't see all parts of a problem at the beginning. Start with the key physics concept, relativity, and work smaller problems as you need them.

• For relativity problems, a useful picture shows the location of the observers and any other information. It is not a good idea to put labels on the information when you first draw the picture. Wait until you determine the proper and/or prime frame so that you can lable the quantities correctly.

• In this case, the observers on the two spaceships each measure the time of transit of the asteroid so this is a time dilation problem regardless of what information is given. Notice, however, that you need to know the relative speed of the two spaceships in order to calculate the time interval, and so at this point you have to recognize that this is a multi-part problem even if you weren't aware of that before.

Your pre-problem, then, requires that you find the relative speed of the ships. If, rather than comparing the time intervals measured by the two astronauts, you instead compare the time interval measured by one astronaut to that measured by Mission Control, you can use the time dilation equation to find the velocity of that astronaut's ship.

• Comparison of time measurements made on the Earth and on the spaceship allows us to find the relative speed of the spaceship and the Earth. We know that the second ship approaches Earth at 1/3 the speed of the first, or (1/3)(0.60 c) =(0.20 c). Scroll down to use this information to find the relative speed between the two spaceships.

----------------------------------------------------------------------------------------------

Step 2

Before you can work with the velocity addition equation, you need to identify who is the prime observer and who 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 on the figure. If you struggle with this step, go to Speed of a Spaceship III for a more detailed example of finding the relative speed between two reference frames.

Now that you know the relative speed between the spaceships, you are ready to return to the original question. Scroll down to find the time of transit of the asteroid as measured by an observer on the second spaceship.

----------------------------------------------------------------------------------------------

Step 3

According to an observer on the second spaceship, it takes 2.8 hours for the asteroid to pass out of sight of the first ship.

• This was a three part problem. You were asked to compare time measurements made by two observers, and so it is a relativisitic time dilation problem. However, you were not given the relative speed between the observers and so had to begin by calculating that speed. The first step was to find the speeds of each of the two spaceships relative to Earth.

Once the speeds of the two ships relative to Earth are known, the velocity addition equation was used to find the relative speed between the ships. At this point, we were able to compare the time measurements between the ships as asked.

As expected, the observer on the second ship measures a longer time interval than the observer on the first ship. As astronaut 2 watches 2.0 hours tick off on a clock in the first ship, he sees that clock run slow so more time ticks off on his clock. The reverse is true as astronaut 1 watches a clock on ship 2. Crazy!