In a tug of war with equal amounts of force applied on both sides of a cart, the cart will not move. This is because the forces are balanced and cancel each other out. In physics, when two opposing forces of equal magnitude act on an object, the net force acting on that object is zero.
According to Newton's first law of motion, an object at rest remains at rest if the net external force acting on it is zero. Since the cart is experiencing equal and opposite forces, it remains stationary.
In summary, the cart stays in place because the forces are balanced and do not result in any net movement.