If we consider ourselves traveling in a bus and if we a throw a ball upwards, the ball should fall behind us according to law of inertia, right? If so why doesn't the ball always fall behind us each time when we throw it upwards? Earth is also moving like bus right?
Assuming no air resistance, the Earth not rotating, the bus moving at constant velocity etc.
If you are looking at the motion of the ball relative to the bus, ie sitting inside the bus, what would you observe?
You would observe the ball going vertically upwards and then coming down vertically downwards.
How would you explain this?
The only force on the ball is its weight which acts vertically downwards and there are no horizontal forces, so if you threw the ball with no horizontal velocity relative to the bus, the ball would continue to have no horizontal velocity as there are no horizontal forces.
What would an observer on the ground see?
That observer would observe you throwing the ball upwards whilst at the same time the ball would have a horizontal velocity equal to that of you and the bus.
Thus the observer on the ground would see you moving at a constant horizontal velocity and the ball moving with the same constant horizontal velocity ie the ball would always be seen by the observer on the ground to be vertically above you and the trajectory of the ball to the observer on the ground would be a parabola - projectile motion.
With air resistance in an open topped bus there would be a horizontal force on the ball in a direction opposite to that of the bus and so the magnitude of the horizontal velocity of the ball would decrease and the ball would progressively lag behind you and return behind you.
Because when you hold the ball, it is accelerated just like you are whether it is in the bus or standing on Earth's surface. Let's take the example of the bus: You, the ball, and the bus are moving at the same speed. So when you throw the ball upwards, it still retains that motion that it had before. As it is only in the air for a short time, it is not deccelerated enough to fall behind you. In short, the motion of the ball stems from you throwing it but also from the motion of the bus before you threw it.
The ball will fall behind you on the bus only if the bus is accelerating. If its speed is constant, the ball will fall right back down. As far as the earth goes, you can consider its velocity as being locally constant and for the same reason the ball does not fall behind.
Hope this helps.
The ball will fall behind you if there is considerable air drag or the bus accelerates while the ball is in the air.
If there is no air drag and the bus moves with constant speed (this includes speed 0) in a straight line and you throw the ball vertically, it will fall back into your hands.
This is because both motions (horizontal and vertical) don't affect each other. The ball always travels horizontally just like you and additionally moves up and down. Because there is nothing that influences the ball's vertical motion, it does the same as you in horizontal direction.