6. The diagram on the left shows a train of mass 50 tonnes approaching a buffer at the end of a straight horizontal railway track. The buffer is designed to prevent the train from running off the end of the track. The buffer may be modelled as a light horizontal spring \(A B\), as shown in the diagram on the right, which is fixed at the end \(A\). The train strikes the buffer so that \(P\) makes contact with \(B\) at \(t = 0\) seconds. While \(P\) is in contact with \(B\), an additional resistive force of \(250000 v \mathrm {~N}\) will oppose the motion of the train, where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of the train at time \(t\) seconds. The spring has natural length 1 m and modulus of elasticity 312500 N . At time \(t\) seconds, the compression of the spring is \(x\) metres.
\includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-7_358_1506_824_283}

1. Show that, while \(P\) is in contact with \(B\), \(x\) satisfies the differential equation $$4 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 20 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 25 x = 0$$
2. Given that, when \(P\) first makes contact with \(B\), the speed of the train is \(U \mathrm {~ms} ^ { - 1 }\), find an expression for \(x\) in terms of \(U\) and \(t\).
3. When the train comes to rest, the compression of the buffer is 0.3 m . Determine the speed of the train when it strikes the buffer.
4. State which type of damping is described by the motion of \(P\). Give a reason for your answer.