What would happen to the Earth, if the Sun suddenly vanishes? (Neglect the gravitational force due to other planets)
The Earth will move straight along the direction of velocity at that time.
The Earth orbits around the Sun because of the gravitational force between the Sun and the Earth. If suddenly the Sun vanishes, the gravitational force between the Sun and the Earth will disappear. So, the Earth will no longer be able to orbit around the Sun. Thus it will start moving in the direction of velocity in a straight line. It will be same as whirling of stone using a thread. If suddenly the thread breaks, the stone will break its path and move in a straight line along the direction of velocity.