How fast must a spaceship travel to reachthe center of the Galaxy (26,000 ly away) in 100 years "ship time"?

To determine how fast a spaceship must travel to reach the center of the Galaxy, 26,000 light-years away, in 100 years "ship time", we need to consider the effects of time dilation. Time dilation occurs due to the theory of relativity, which states that time moves slower for an object in motion relative to another object at rest.

To calculate the required speed, we can use the equation:

v = d / t

where:
v is the velocity of the spaceship,
d is the distance to be traveled (26,000 light-years), and
t is the time taken (100 years "ship time").

However, since the spaceship is moving at a significant fraction of the speed of light, we need to account for relativistic effects. According to special relativity, time dilation occurs as velocity approaches the speed of light (c, approximately 3 x 10^8 meters per second).

The formula to calculate time dilation is:

t' = t / √(1 - (v^2 / c^2))

where:
t' is the time experienced by the moving spaceship,
t is the time measured by an observer in the stationary frame (100 years), and
v is the velocity of the spaceship.

Now, we need to rearrange the time dilation formula to solve for velocity:

v = √((1 - (t / t')^2) * c^2)

Since we want to find the velocity needed to travel a distance of 26,000 light-years in 100 years "ship time", we substitute the values into the equation:

t = 100 years
d = 26,000 light-years
c = 9.461 x 10^15 meters (speed of light)

Now, let's plug in the values and calculate the velocity:

v = √((1 - (t / t')^2) * c^2)
v = √((1 - (100 / d)^2) * (9.461 x 10^15)^2)

After calculating this equation, we will get the velocity required for the spaceship to reach the center of the Galaxy in 100 years "ship time".