(1) The speed with which the star's light will pass you is 3 x 10⁸ m/s.
(2) The speed of the other spaceship is -2.943 x 10⁸ m/s.
The given parameters;
- speed of the start light, u' = c
- speed of spaceship, v = 0.8c
(1)
Since you are travelling away from the star (same direction), relative to a third observer, the speed at which the star's light will pass you is calculated using Lorentz velocity addition.
[tex]u = \frac{u'+ v}{1 + \frac{v}{c^2} u'} \\\\u = \frac{c+(-0.8c)}{1 + \frac{-0.8c}{c^2} (c)}\\\\u = \frac{c-0.8c}{1- \frac{0.8c}{c} } \\\\u = \frac{0.2c}{0.2} = c = 3\times 10^{8} \ m/s[/tex]
Thus, the speed with which the star's light will pass you is 3 x 10⁸ m/s.
(2) The speed of the spaceship approaching = 0.84 c
Since you are travelling in opposite direction to second spaceship, relative to a third observer (frame of reference), the speed of the other spaceship is calculated using Lorentz velocity addition.
- let the speed of the approaching spaceship u', = 0.84c
[tex]u = \frac{u' + v}{1 + \frac{v}{c^2}u' } \\\\u = \frac{-0.84c - 0.8c}{1 + \frac{(-0.8c)}{c^2}(-0.84c) } \\\\u = \frac{-1.64c}{1.672} = -0.981c \\\\u = -0.981 \times 3\times 10^{8} = -2.943 \times 10^{8} \ m/s[/tex]
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