Yes, the crews of the two spaceships can use the observed time dilation to calculate the relative velocity (v) between the two spaceships using the Lorentz time dilation formula. The formula relates the observed time dilation to the relative velocity between the two reference frames.
The Lorentz time dilation formula is given by:
t' = t / √(1 - (v^2/c^2))
Where: t' is the observed time interval (or time dilation) for an observer moving relative to the stationary observer. t is the proper time interval (time measured by the stationary observer). v is the relative velocity between the two reference frames. c is the speed of light in a vacuum (approximately 299,792,458 meters per second).
To calculate the relative velocity (v), you need to rearrange the formula:
v = c √(1 - (t'^2/t^2))
Given the observed time dilation values (t'1, t'2) measured by the crews of the two spaceships, and the proper time interval (t), you can substitute these values into the formula to calculate the relative velocity (v) between the two spaceships.
It's important to note that this formula gives the relative velocity between the two spaceships in the frame of reference of the observer measuring the time dilation. If the two spaceships are moving relative to a third observer or reference frame, then the velocity between the two spaceships would be different when measured from that reference frame.