Free ^ 238 U nuclei placed inside a train emit alpha particles....

The train moving at a uniform speed $v$ represents an inertial frame of reference.

According to the principle of Galilean relativity, the laws of physics and the internal dynamics of a system remain the same in all inertial frames.

Since the ^{238}\text{U}$nucleus is placed inside the train, it is initially at rest with respect to the passenger in both cases. The velocities of the emitted alpha particle and the recoiling nucleus relative to the passenger will be identical to those in the stationary case.<br><br>Alternatively, if$\vec{v}_{\alpha}$and$\vec{v}_{R}$are the velocities of the alpha particle and the recoiling nucleus in the train's frame, the separation at time$t$is$|\vec{v}_{\alpha} - \vec{v}_{R}|t = x$.<br><br>Therefore, the separation between the alpha particle and the recoiling nucleus at time$t$, as measured by the passenger, remains$x.