A cart moves along a level, horizontal track with a constant acceleration \(a\) directed in the \(+x\)-direction. At time \(t = 0\), the cart passes the origin \(x = 0\) with velocity \(v_0 \hat{i}\), and simultaneously launches a small ball vertically upward with speed \(u_0\) relative to the cart. Air resistance is negligible, and the downward acceleration due to gravity is \(g\). Which of the following correctly gives the trajectory equation \(y(x)\) of the ball in the ground reference frame (for \(y \ge 0\)) and the position of the ball relative to the launcher, \(\Delta x = x_{\text{ball}} - x_{\text{cart}}\), at the instant the ball returns to its launch height?