An open cart moves along a straight, horizontal track aligned with the \(x\)-axis with a constant acceleration \(\vec{a} = a_0 \hat{i}\). At time \(t = 0\), the cart is at rest at the origin \((x = 0, y = 0)\) of an inertial reference frame. At that same instant, a small projectile is launched from the origin with an initial velocity \(\vec{v}_0 = v_0 \hat{j}\). Let the cart define a non-inertial reference frame with coordinates \((x', y')\), whose origin is fixed to the cart and is at \((0,0)\) at \(t = 0\). Neglecting air resistance, which of the following expressions correctly describes the vertical position \(y'\) of the projectile as a function of its horizontal position \(x'\) in the reference frame of the cart throughout its flight?