A spaceship from another galaxy passes over the solar system directly above and

Question

A spaceship from another galaxy passes over the solar system directly above and parallel to a radial line from the Sun to the Earth. We, on Earth, measure the Earth-Sun distance to be 1.50 times 10^11 m and measure the speed of the spaceship to be 0.8c. (a) What docs a scientist in the spaceship measure the Earth-Sun distance to be? (b) How long does the trip take according to a clock in the spaceship? In the TV series, Star Trek, the Next Generation, Captain Picard and his crew often encountered the dreaded Borg who were cybernetic beings bent on conquering the galaxy by assimilating civilizations into the Borg collective. A Borg spaceship is typically a cube of 100 m on a side with a mass of 1.00 times 10^9 kg. (a) If a Borg spaceship is determined to be traveling at 0 9c relative to the Earth with one face parallel to the x-axis, what is the mass of this vessel measured by observers at rest on the Earth? (See the diagram) (b) What arc the dimensions of the Borg spaceship as measured by observers at rest on Earth? (c) What is the average density of a Borg spaceship as measured by observers on Earth?

Explanation / Answer

part a:

due to length contraction, distance measured by scientist on the spaceship is

actual length*sqrt(1-(v/c)^2)

where v=speed of the space ship as observed from earth

=1.5*10^11*sqrt(1-0.8^2)

=9*10^10 m

part b:

time taken=distance measured/speed

=9*10^10/(0.8*3*10^8)

=375 seconds

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