The speed of the spaceship is 0.971c
Explanation:
Since Steve is moving at a speed close to the speed of light, the time observed by Kasey will be dilated, according to the equation
[tex]T' = \frac{T}{\sqrt{1-(\frac{v}{c})^2}}[/tex]
where
T' is the time measured by Kasey
T is the time measured by Steve
v is Steve's speed
c is the speed of light
Here we have:
T = 10 years (time measured by Steve)
When Steve is back, his age is 48 + 10 = 58 years. Kasey has now the same age, so the amount of time passed according to Kasey is
[tex]T' = 58 -16=42 y[/tex]
Substituting into the equation, we can fidn the speed of Steve's spaceship:
[tex]\sqrt{1-(\frac{v}{c})^2}=\frac{T'}{T}\\1-(\frac{v}{c})^2=(\frac{T'}{T})^2\\v=c\sqrt{1-(\frac{T'}{T})^2}=c\sqrt{1-(\frac{10}{42})^2}=0.971 c[/tex]
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