Question

(Include full solutions)You and your friend Paul want to have a bet on how fast it can take to reach another friend Peter who is near the star Sirius B. Knowing that the star has a proper length of 8.6 light years from Earth, Paul thinks that since the maximum speed of your ship is 70% that of the speed of light, it will take about 9 years to get there. However, you reason that around 12 years will make more sense.

a)      Who is right from the perspective of you and Paul?

b)      When you reach Peter and explain the bet, he disagrees about who was right, why is that?

Answer 1

we have,

maximum speed of our spaceship ,

where, speed of light in vacuum

the velocities of celestial bodies earth and star Sirius are so small comparative to the speed of light so we will assume them as stationary with respect to each other,

Peter is near the star Sirius and is stationary with respect to star Sirius,

therefore,

frame of reference of peter = frame reference of star Sirius and Earth

we have,

proper distance between between star Sirius A and earth light years

Here, proper length ( distance ) between star Sirius and earth corresponds distance in the frame of reference of Peter ( which as mentioned earlier equivalent to frame of reference of star Sirius and Earth )

  

1 )

We and Paul are moving on a spaceship with the speed of   from earth to star Sirius.

So the lorentz factor will be given as,

So as per the lorentz contraction , the distance measured between star Sirius and earth in our frame of reference ( frame of reference of us and Paul ) will be given by fallowing relation as,

  

so,

using and    we get,

Let,

time taken by our spaceship to fly from earth to Sirius in our frame of reference

It will be given as,

So, time measure in our frame of reference ( Expressed in one significant figures )

So from the perspective of us and Paul that is in our frame of reference time taken by spaceship is  .

So we see that from our perspective Paul is right.

2)

Let,

time taken by our spaceship to fly from earth to Sirius in Peter's frame of reference

It will be given as,

So, time measure in Peter's frame of reference ( Expressed in two significant figures )

that is, time taken by our spaceship to reach Peter from earth as measured by peter  

So, that's why Peter disagrees with who was right ( answer to previous part )

( Note :- In terms of physics of special theory of relativity we and Paul both are right, the difference in opinions was because of different frame of reference we were considering for time measurement,

When we meet Peter, for only 9 years will have passed and for Peter he had wait for 12 years , so in terms of biological age Peter aged by 12 years and we aged by 9 years ! )