Answer: 0.99973 c
Explanation:
Betelgeuse is 430 ly away from earth = 4.07 ×10¹⁸ m
You want to age only 20 years during the round trip. It means you want to cover one way distance in 10 years relative to Earth.
10 years = 3.15 × 10⁸ s
This is possible only when rocket travels close to speed of the light.
We will use time dilation formula:
\(t_o=t' \sqrt {1-\frac{v^2}{c^2}}\)
where, \(t_o\) is the time elapsed in Earth's frame of reference, t' is the time elapsed in rocket's frame of reference, v is the velocity of the rocket and c is the speed of light.
\(t_o=\frac{d}{v} \sqrt {1-\frac{v^2}{c^2}}\)
where d is the distance of Betelgeuse from Earth.
\(\Rightarrow t_o\times v=d \sqrt {1-\frac{v^2}{c^2}}\)
\(\Rightarrow t_o^2v^2=d^2-\frac{d^2v^2}{c^2}\)
\((t_o^2+\frac{d^2}{c^2})v^2=d^2\\ \Rightarrow v=\frac{d}{\sqrt {t_o^2+\frac{d^2}{c^2}}}\)
\(\Rightarrow v = \frac{4.07\times10^{18} m}{\sqrt{(3.15\times10^8 s)^2+\frac{(4.07\times 10^{18}m)^2}{(3\times 10^8m/s)^2}}}=\frac{4.07\times 10^{18} m}{1.36\times 10^{10}s}=2.99\times10^8m/s = 0.99973 c\)
