Question

You are traveling in a spaceship toward a distant colony when you receive a distress signal from the colony. The signal is at a frequency of 4.0 GHz, while the usual distress band is 2.0 GHz. The message says that an alien attack is imminent, and when the signal was sent they estimated that the invaders would arrive in 1.5 hours local time. It will take you 1 hour ship time to get there. Will you be in time to help with the colony defense? By how much will you be late or early?

Answer 1

In order to answer this question, the Doppler equation for light is needed, as well as the equation for relativisitic time dilation.

In the Doppler equation for light, the observed frequency f_o is related to the frequency of the source f_s by:

with v being the velocity of the source, and c being the speed of light.

The equation for relativistic time dilation is:

with t_o being the observed change in time, t_s being the change in time at the source, v being the velocity of the source, and c being the speed of light.

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Step 1) To find if the person arrives in time to help the colony or not, their speed needs to be found, and then used in the time dilation equation to see how long the hour for him takes for a stationary observer. So with the information about the frequencies, the Doppler equation needs to be used to find the velocity of the ship.

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Step 2) Start by multiplying both sides of the Doppler equation by the denominator, and squaring both sides to begin to isolate the velocity:

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Step 3) Distribute the frequencies:

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Step 4) Get all the v/c terms together on one side:

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Step 5) Isolate the v/c terms:

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Step 6) Evaluate the ratio by using 2.0 GHz for the source frequency f_s, and 4.0 GHz for the observed frequency f_o:

So the ship is going at 60% the speed of light.

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Step 7) Evaluate how much time passes for a stationary observer by using 1 hr for the change in time of the source t_s in the time dilation equation, and 0.6 for the ratio v/c:

So a stationary observer sees that the ship will take 1.25 hours to reach the colony. Since the colony was expecting the attack to occur in 1.5 hours, the ship will arrive there 0.25 hours (or 15 minutes) early.