She invites you to go on a long journey with her into space in her small spacesh

Question

She invites you to go on a long journey with her into space in her small spaceship of
mass 1000kg. Using thermodynamics approach, she tells you that stellar bodies can
be assumed to have an average mass of 1030kg ( mass of the sun ), and to move with
a random velocity of 10 km/s. She wants to know the average velocity the rocket
will reach after a very long time

hint here

When you put a bunch of particles together they interact with each other until they get to an equilibrium. At equilibrium they all begin to have the same kinetic energy, so in our problem, the spaceship and any star in space will have the same kinetic energy. Since the star has a bigger mass, its kinetic energy will be the classic one that you know, which is 1/2mv^2, but the space ship is light so it will have a relativistic kinetic energy. So you need to set them equal to each other. The relativistic kinetic energy. It is easy to find it. It is in your textbook way in the back. I hope this will help.

Explanation / Answer

since we need to consider the relativistic mass of the spaceship,

the mass of the spaceship will be mo/(1-(v/c)^2)^0.5

so equating the kinetic energies,

0.5*10^30*(10^4)^2=0.5*1000*v^2/(1-(v^2/(3*10^8)^2))^0.5

solving, we get

v=2.9999*10^8

or v~3*10^8 (i.e. they need to travel at the speed of light)

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