Sirius is about 9.0 light-years from Earth.

To reach the star by spaceship in 10 years (ship time), how fast must you travel?
How long would the trip take according to an Earth-based observer?
How far is the trip according to you?

To calculate the speed required to reach Sirius in 10 years of ship time, we need to apply the concept of time dilation, as predicted by Einstein's theory of relativity.

Firstly, we need to understand that as an object approaches the speed of light, time dilation occurs, which means that time moves slower for the moving object relative to a stationary observer. The faster the object travels, the more time dilation occurs.

To find the speed needed to reach Sirius in 10 years of ship time, we will use the time dilation formula:

t(ship) = t(earth) / √(1 - (v^2 / c^2))

where:
t(ship) is the ship time (10 years in this case),
t(earth) is the time measured by the Earth observer,
v is the velocity of the spaceship, and
c is the speed of light.

We can rearrange this formula to solve for v:

v = c * √(1 - (t(earth) / t(ship))^2)

Given that t(ship) is 10 years, we can substitute this value into the equation:

v = c * √(1 - (t(earth) / 10)^2)

Calculating the value of v will give us the required velocity to reach Sirius in 10 years of ship time.

To find how long the trip would take according to an Earth-based observer, we can use the concept of time dilation in reverse. According to the Earth-based observer, who is not moving relative to Earth, time appears normal. Therefore, the trip duration from their perspective will be the same as the distance divided by the speed of light.

The distance to Sirius is given as 9.0 light-years. So, the trip duration according to an Earth-based observer can be calculated as:

t(earth) = distance / c

Calculating this will give us the time it takes according to an Earth-based observer.

Finally, to find the distance of the trip according to the spaceship, we multiply the ship's velocity by the ship time:

distance(ship) = v * t(ship)

Calculating this will give us the distance as perceived by someone on the spaceship.

Note: In the calculations, we are assuming constant velocity, which may not be practical for actual space travel where acceleration and deceleration need to be taken into account. This is a simplified approach to demonstrate the concepts involved.