Question

The nearest star to Earth is Proxima Centauri, 4.2 light-years away.

Part A

At what constant velocity must a spacecraft travel from Earth if it is to reach the star in 3.9 years, as measured by travelers on the spacecraft?

Express your answer using two significant figures.

v=_____c

Part B

How long does the trip take according to Earth observers?

Express your answer using two significant figures.

t=_____ years

Answer 1

distance between nearest star to earth, D = 4.2 light year

Part-A :

actual time = observer time x $\gamma$

$\gamma$ = 1 / _$\sqrt{}$(1 - V²/c² )                                     { eq. 1 }

where, $\gamma$ = relativistic time contraction factor

speed, V = distance / actual time

V = distance / observer time x $\gamma$                           { eq. 2 }

where, observer time = 3.9 year

inserting the values in eq.2,

V = (4.2 LY) / (3.9 year x $\gamma$)                                { eq. 3 }

V is measured in LY/Y or V/c

V / c = 1.0769 (1 / _$\gamma$)

V / c = 1.0769 [ _$\sqrt{}$(1 - V²/c² )                             { eq. 4 }

squaring on both sides,

V² / c² = 1.1597 (1 - V²/c² )   

V² / c² = 1.1597 - 1.1597 (V² / c²)

2.1597 (V² / c²) = 1.1597

(V² / c²) = 0.5369

V / c = 0.7327

V = 0.73 c               (two significant figure)

or V = 0.73 (3 x 10⁸ m/s) = 2.19 x 10⁸ m/s

Part-B :

Part-B : according to the observer on earth, the spacecraft moves at a speed of (0.73 c = 0.73 light-years per year).

the time for the trip is given as :

t = D / V

where, D = distance    &    V = speed

so, t = (4.2 light year) / (0.73 light years per years)

t = 5.75 years