A particle \(P\) of mass \(0.5\,\text{kg}\) moves in a straight line. At time \(t\) seconds, the velocity of \(P\) is \(v\,\text{m s}^{-1}\), and its displacement from a fixed point \(O\) on the line is \(x\) metres.

The only forces acting on \(P\) are a force of magnitude \(\dfrac{150}{(x+1)^2}\,\text{N}\) in the direction of increasing displacement and a resistive force of magnitude \(\dfrac{450}{(x+1)^3}\,\text{N}\).

When \(t=0\), \(x=0\) and \(v=20\).

Find \(v\) in terms of \(x\), giving your answer in the form \(v=\dfrac{Ax+B}{x+1}\), where \(A\) and \(B\) are constants to be determined.