Relativity : Traveling to Andromeda galaxy

[Delzac]

Homework Statement

An astronaut wishes to visit the Andromeda galaxy 2 million light years from Earth. He wishes the one way trip to take him 30 years (ie in the frame of reference of the spaceship). Assuming that his speed is constant, how fast must he travel?

Homework Equations

$$L = L_0/\gamma$$
$$\Delta T = \gamma \Delta T_0$$

The Attempt at a Solution

I know that by increasing velocity of the spacecraft , you can effectively length contract the distance between Earth and the galaxy, however, i do not know how i should equate the equations. Any help will be greatly appreciated.

 

[Delzac]

Ermm...any help?

 

[m2003]

I would approach this problem by first considering the concept of special relativity, which states that the laws of physics are the same for all observers in uniform motion. This means that the astronaut's frame of reference, the spaceship, and the frame of reference of Earth can be considered equivalent.

To solve for the required speed, we can use the equation L = L_0/\gamma, where L_0 is the distance between Earth and Andromeda (2 million light years) and L is the length contracted distance in the frame of reference of the spaceship. We can also use the equation \Delta T = \gamma \Delta T_0, where \Delta T_0 is the time it takes for light to travel from Earth to Andromeda (2 million years) and \Delta T is the time it takes for the astronaut to travel to Andromeda in the frame of reference of the spaceship (30 years).

Substituting the given values, we get:

L = 2 million light years / \gamma
\Delta T = 30 years = \gamma * 2 million years

Solving for \gamma in the second equation, we get \gamma = 30/2 million = 1.5*10^-5. Substituting this value into the first equation, we get:

L = 2 million light years / (1.5*10^-5) = 133.33 light years

This means that in the frame of reference of the spaceship, the distance between Earth and Andromeda is only 133.33 light years. To find the required speed, we can use the formula v = d/t, where d is the distance and t is the time. Substituting the values, we get v = 133.33 light years / 30 years = 4.44 light years per year.

Therefore, the astronaut must travel at a speed of 4.44 light years per year to reach the Andromeda galaxy in 30 years in the frame of reference of the spaceship. This speed is incredibly fast, and it shows the immense distances and speeds involved in traveling to other galaxies.