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You and a friend pass each other in spaceships that are each traveling 0.15c relative to...

You and a friend pass each other in spaceships that are each traveling 0.15c relative to the other. Your friend takes a nap and you decide to time how long your friend's nap takes. In your frame, your friend's nap takes 3.5 h. How long was your friend's nap in their frame? Unbeknownst to you, another person on your friend's ship saw you timing your friend's nap and decided to time how long you were timing your friend's nap. In that person's frame (which is the same as your friend's frame), how long were you timing your friend's nap?

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Answer #1

Time dilation

is time interval of spaceship moving frame with velocity .

is proper time interval of spaceship measured in rest frame.

In the observer's frame(in the first spaceship)

The time interval of friend's nap in second spaceship moving with velocity is  

Time interval of friend's nap in their frame is

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In friend's frame of reference:

Time interval of your measurement of friend's nap in rest frame

Time interval of timing measured by a person in friend's frame is

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