A particle of mass 0.5 kg moves along a horizontal straight line. Its velocity is \(v \mathrm{~m} \mathrm{~s}^{-1}\) at time \(t \mathrm{~s}\). The forces acting on the particle are a driving force of magnitude 50 N and a resistance of magnitude \(2 v^{2} \mathrm{~N}\). The initial velocity of the particle is \(3 \mathrm{~ms}^{-1}\).
(a) Find an expression for \(v\) in terms of \(t\).

(b) Deduce the limiting value of \(v\).