A cart of initial mass \(m_0\) moves along a straight, frictionless horizontal track at a constant speed \(v_0\). Rain falling vertically enters the open top of the cart such that the total mass of the cart and accumulated water increases with time according to \(m(t) = m_0 + b t^2\), where \(b\) is a positive constant. An engine exerts a horizontal force \(F(t)\) on the cart to maintain its constant speed. Which of the following expressions represents the magnitude of the horizontal force \(F(t)\) as a function of time \(t\)?