Barnard\'s Star, a nearby star in our galaxy, is 9.54 light-years away. This mea

Question

Barnard's Star, a nearby star in our galaxy, is 9.54 light-years away. This means that, as measured by a person on Earth, it would take light 9.54 years to reach this star. A rocket leaves for Barnard's Star at a speed of v = 0.95c relative to the Earth. Assume that the Earth and Barnard's Star are stationary with respect to one another. Work out the distance to Barnard's Star, as measured by Earth observers, in km? Referring to the distance calculated in part (a), is it a proper distance? Justify your answer. According to the astronauts, how much did they age (in years) during their journey? According to the Earth observers, how much time has passed during the astronauts' journey? According to the astronauts, how far (in km) did they travel? One of the astronauts, using a meter stick, measures the length and diameter of the cylindrical spacecraft to be 82 and 21m, respectively. Assuming that the spacecraft is oriented with its long cylindrical axis in the direction of motion, what are the dimensions of the spacecraft, as measured by an observer on Earth?

Explanation / Answer

Here,

a) distance = 9.54 light years

distance = 9.54 * (3 *10^5 km) * 3600 * 24 * 365

distance = 9.026 *10^13 km

the distance measured by earth's observer is 9.026 *10^13 km

b)

Yes, it is a proper distance , as the observer on earth is an inertial frame of reference

c)

let the time taken by astronauts is t

time = distance/speed

time = 9.54 * sqrt(1 - (v/c)^2)/v

time = 9.54 * sqrt(1 - 0.95^2)/.95

time = 3.13 years

the time taken to travel this distance is 3.13 years as measured by the astronauts

d)

for the time according to those present at earth

time according to those present at earth = 3.13/sqrt(1 - (v/c)^2)

time according to those present at earth = 3.13/sqrt(1 - 0.95^2)

time according to those present at earth = 10.02 years

the time according to those present at earth taken is 10.02 years

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