Two carts, Cart 1 and Cart 2, each of mass \(m\), move in the positive \(x\)-direction along a horizontal frictionless track with initial speeds \(v_0\) and \(3v_0\), respectively. Starting at time \(t = 0\), both carts are subjected to the same time-dependent horizontal force \(F(t) = c t\) in the positive \(x\)-direction until time \(t = T\), where \(c = \dfrac{4mv_0}{T^2}\). What is the ratio \(\dfrac{\Delta K_1}{\Delta K_2}\) of the change in kinetic energy of Cart 1 to the change in kinetic energy of Cart 2 over this time interval?