A small cart of mass \(m\) is constrained to move along a straight, horizontal, frictionless track that aligns with the positive x-axis. The cart is initially at rest at the origin \(x = 0\). A stationary magnetic launcher located at the origin exerts a repulsive force on the cart in the positive x-direction. The magnitude of this force as a function of position \(x\) is given by \(F(x) = F_0 e^{-\alpha x}\), where \(F_0\) and \(\alpha\) are positive constants.