Question

An astronaut must journey to a distant planet, which is 211 light-years from Earth. What speed will be necessary if the astronaut wishes to age only 15 years during the trip? (Give your answer accurater to five decimal places.) Hint: The astronaut will be traveling at very close to the speed of light. Therefore, approximate the dilated trip time At to be 211 years

Answer 1

• The speed necessary is 0.99747 c

Step-by-step explanation:

We know that the equation for time dilation will be:

`\Delta t = \frac{\Delta t'}{\sqrt{1-(v^2)/(c^2)}}`

where Δt its the time difference measured from Earth, and Δt' is the time difference measured by the astronaut.

Lets work a little the equation

`\sqrt{1-(v^2)/(c^2)} = (\Delta t')/(\Delta t)`

`1-(v^2)/(c^2)= ((\Delta t')/(\Delta t))^2`

`(v^2)/(c^2)= 1 - ((\Delta t')/(\Delta t))^2`

`(v)/(c)= \sqrt{ 1 - ((\Delta t')/(\Delta t))^2 }`

`v = \sqrt{ 1 - ((\Delta t')/(\Delta t))^2 } c`

So, we got our equation. Knowing that Δt=211 years and Δt'=15 years

then

`v = \sqrt{ 1 - ((15 \ y)/(211 \ y))^2 } c`

`v = 0.99747 c`