Time dilation spacecraft travel problem

[Benzoate]

Problem: How fast must a spacecraft travel relative to the Earth for each day on the Earth to correspond to 2 days on the earth?

My solution:

proper time = 1 day = 86400 seconds
time on space craft= 2 days= 172800 seconds
relavant equation: t=t(0)/(sqrt(1-v^2/c^2)

I would plug in both the proper time and relative time into time dilation equation in order to find the speed of the spacecraft correct?

 

[G01]

The time on the spacecraft would be the proper time. Other than this you are correct.

The proper time is the time between events measure by a clock that is present and at rest at both events. Here, the events are the two points mentioned in the spacecraft 's journey. The only clock present and at rest at both of these events is the one on the ship.

 

[m2003]

Yes, you are on the right track. In order to find the speed of the spacecraft, we can rearrange the time dilation equation to solve for the velocity (v). The equation would look like this:

v = c * sqrt(1 - (t / t(0))^2)

Where c is the speed of light and t is the time on the spacecraft (2 days or 172800 seconds). We can then plug in the proper time (1 day or 86400 seconds) to find the velocity needed for the spacecraft to experience time dilation and have 2 days on Earth correspond to 1 day on the spacecraft. To ensure accuracy, it is important to use the same units for time (seconds) and speed (meters per second).