A free ^ 238 U nucleus kept inside a train emits an alpha particle....

The separation between the alpha particle and the recoiling nucleus depends only on their relative velocity.

Let the velocity of the alpha particle be $\vec{v}_{\alpha}$ and that of the recoiling nucleus be $\vec{v}_{N}$ in the frame of the train. The relative velocity is $\vec{v}_{\text{rel}} = \vec{v}_{\alpha} - \vec{v}_{N}$.

Since the train moves with a uniform velocity $v$, it represents an inertial frame of reference. According to the principle of relativity, the laws of physics are identical in all inertial frames. The energy released in the decay and the conservation of momentum in the train's frame remain exactly the same as when the train was stationary.

Thus, the relative velocity $\vec{v}_{\text{rel}}$ between the alpha particle and the recoiling nucleus is independent of the uniform velocity of the train. The separation at time $t$, given by $x = |\vec{v}_{\text{rel}}|t$, will remain unchanged.

Therefore, the distance measured by the passenger at time $t$ is $x$.