According to time dilation, the elapsed time between two events is different for observers in relative motion to each other. The formula for time dilation is given by:
Δt' = Δt / sqrt(1 - v^2/c^2)
where Δt is the time interval measured by the observer in the rest frame, Δt' is the time interval measured by the observer in the moving frame, v is the relative velocity between the two frames, and c is the speed of light.
In this problem, the neutron is at rest relative to the first observer and its lifetime is measured to be 900 s. The neutron is moving relative to the second observer, and its lifetime is measured to be 2065 s. We want to find the velocity v of the neutron relative to the second observer.
We can use the time dilation formula twice, once for each observer:
Δt' = Δt / sqrt(1 - v^2/c^2)
For the first observer (rest frame of the neutron), Δt = 900 s and Δt' = 2065 s (measured by the second observer):
2065 = 900 / sqrt(1 - v^2/c^2)
Simplifying and rearranging:
(1 - v^2/c^2) = (900/2065)^2
1 - v^2/c^2 = 0.155
v^2/c^2 = 0.845
v/c = sqrt(0.845)
v = 0.919c
Therefore, the neutron is moving at a speed of 0.919 times the speed of light relative to the second observer.