How Fast Must a Spaceship Travel to Cover 12 Light-Years in 7 Years?

[arp777]

Homework Statement

The distance from Planet X to a nearby star is 12 Light-Years (a light year is the distance light travels in 1 year as measured in the rest frame of Planet X).

(A) How fast must a spaceship travel from Planet X to the star in order to reach the star in 7 years according to a clock fixed on the spaceship?

(B) How long would the trip take according to a clock fixed on Planet X?

(C) What is the distance from Planet X to the nearby star, according to an astronaut on the spaceship?

Homework Equations

Time-dilation and length-contraction equations.

The Attempt at a Solution

(A) Using the only numbers the problem statement provides so far, I have a distance, and a change in time. Using the distance between planet x and the star (12 light years) and the time interval (7 years), obviously I end up with a number that is greater than the speed of light. v = (distance)/(time) = 1.713 C

This is my speed bump. I don't believe I can approach the rest of the problem using this as my spaceship's velocity, seeing as all relevant equations become imaginary. I must be missing a key point as far as reference frames go and in getting the appropriate velocity of the spaceship, so what am I doing wrong from get go?

 

[Doc Al]

(A) Using the only numbers the problem statement provides so far, I have a distance, and a change in time. Using the distance between planet x and the star (12 light years) and the time interval (7 years), obviously I end up with a number that is greater than the speed of light. v = (distance)/(time) = 1.713 C

Careful. Don't mix distance measured in one frame (the planet X frame) with time measured in a different frame (the spaceship frame).

Give it another shot.

 

[arp777]

I see. So if I'm thinking solely in the spaceship's rest frame, I use the equation that describes time difference in a moving frame(S') relative to a rest frame (S):

i.e. Δt = $\gamma$Δt'

and Δt' = L$_{o}$(v/(c^2))

where Δt' = 7 years and v = velocity of the spaceship in it's own rest frame.

Is this correct? It gives me an appropriate answer, being less than C.

 

[K^2]

Velocity of a spaceship in its own rest frame is always zero. You can either talk about velocity of the ship in frame of Planet X, or velocity of Planet X in frame of the ship. These will be the same except for direction, of course.

While trying to find necessary velocity, the choice of frame is yours. You can either pick Planet X's frame, and say that ship's time is dilated, so 7 years on board will give you longer journey, or you can pick ship's frame and say that distance is contracted and you travel a lot less than 12ly in these 7 years. Either way, when you solve for velocity, you should get the same answer. Might as well check yourself by trying it both ways.

I'm not sure where the factor c^2 comes from in this, "Δt' = L_o(v/(c^2))", but otherwise, you seem to be on the right track for solving this in planet's frame.

 

[Doc Al]

I see. So if I'm thinking solely in the spaceship's rest frame, I use the equation that describes time difference in a moving frame(S') relative to a rest frame (S):

i.e. Δt = $\gamma$Δt'

Good. Δt will give you the travel time in the planet X frame.

and Δt' = L$_{o}$(v/(c^2))

Wrong formula! That describes the amount of clock desynchronization between clocks in the planet X frame (imagine clocks on planet X and on the star) according to the ship frame. You won't need that.

where Δt' = 7 years and v = velocity of the spaceship in it's own rest frame.

v is the velocity of the spaceship with respect to planet X, and vice versa. (As pointed out, the velocity of the ship in its own frame is of course zero.)

Now write an expression for ship's velocity using only measurements from the planet X frame. (Or planet X's velocity from the ship's frame. Your choice.) Then you can solve for v.