Special Relativity (speed taken to age only 13 years)

[Dave218]

An astronaut must journey to a distant planet, which is 186 light-years from Earth. What speed will be necessary if the astronaut wishes to age only 13 years during the round trip? (Give your answer accurate to four decimal places.)

_________c

13y = to 186ly = t

to/t = (√(1-(v^2 / c^2)))^2

(13/186)^2 = 1-(v^2 - c^2)

c^2 x (-(13/186)^2 + 1) = v^2

(3x10^8)^2 x (-(13/186)^2 + 1 ) = v^2

v= 2.99266359x10^8
v=.9975545309c
v=.9976c

this is my attempt but the answer is incorrect

 

[jedishrfu]

You need to use the time dilation formula:

ta = te / sqrt(1 - v^2/c^2) where ta=astronaut age=13

and te=earth time to travel = 186ly / v

where v is the speed of the spaceship which you don't know.

Solve for v.Does that give the right result?

 

[vela]

An astronaut must journey to a distant planet, which is 186 light-years from Earth. What speed will be necessary if the astronaut wishes to age only 13 years during the round trip? (Give your answer accurate to four decimal places.)

_________c

13y = to 186ly = t

to/t = (√(1-(v^2 / c^2)))^2

(13/186)^2 = 1-(v^2 - c^2)

c^2 x (-(13/186)^2 + 1) = v^2

(3x10^8)^2 x (-(13/186)^2 + 1 ) = v^2

v= 2.99266359x10^8
v=.9975545309c
v=.9976c

this is my attempt but the answer is incorrect

Your mistake is in assuming it will take 186 years to get to the planet from Earth as measured in the Earth's rest frame. It will take longer than that because the ship can't move at the speed of light.