Question

Earth's neighboring galaxy, the Andromeda Galaxy, is a distance of 2.54×107 light-years from Earth. If the lifetime of a human is taken to be 90.0 years, a spaceship would need to achieve some minimum speed vmin to deliver a living human being to this galaxy. How close to the speed of light would this minimum speed be? Express your answer as the difference between vmin and the speed of light c.

Answer 1

`0.0018833\ \text{m/s}`

Step-by-step explanation:

`d` = Distance of Andromeda Galaxy from Earth =
`2.54* 10^7\ \text{ly}`

`t` = Time taken =
`90\ \text{years}`

`c` = Speed of light =
`3* 10^8\ \text{m/s}`

We have the relation

`t=t_o\sqrt{1-(v^2)/(c^2)}\\\Rightarrow 90=2.54* 10^7\sqrt{1-(v^2)/(c^2)}\\\Rightarrow (90^2)/((2.54* 10^7)^2)=1-(v^2)/(c^2)\\\Rightarrow 1-(90^2)/((2.54* 10^7)^2)=(v^2)/(c^2)\\\Rightarrow v=c\sqrt{1-(90^2)/((2.54* 10^7)^2)}`

`c-v=c(1-\sqrt{1-(90^2)/((2.54* 10^7)^2)})\\\Rightarrow c-v=3* 10^8(1-\sqrt{1-(90^2)/((2.54* 10^7)^2)})\\\Rightarrow c-v=0.0018833\ \text{m/s}`

The required answer is
`0.0018833\ \text{m/s}`.