I love this problem! I have asked myself this
question many times. I think the best way to think
about this problem is using forces.
According to physics, objects move (or don't
move) because there are forces on them. For
example, when you fire an elastic band, you are
using a spring force to cause the elastic band to
suddenly move. When a ball bounces off the floor,
it is using the normal force from the floor to
suddenly reverse its momentum (plus spring forces
inside the ball--look at this
slow-motion video to see what I mean).
When you are standing still, the gravitational
force pushing you downwards is exactly balanced by
the normal force from the ground pushing you back
up, so you don't move at all. So for the school
bus problem, if we can keep track of all the
important forces on the hovering object, we
can figure out how it will move.
First, let's do a simpler case. What forces
are there on an object in the bus that is not
hovering, say, a student in their seat? As the
bus is beginning to move, the student has no
forces on them that also makes them move. All they
have is gravitational force (down) balanced with
normal force (up). But then the seat, part of the
moving bus, begins pushing into the student's
back. This is also a normal force, but this time
it pushes the student forwards. If the bus
continues to accelerate, the seat will continue to
push the student forwards with a normal force.
This way the student "keeps up" with the bus.
Now:what forces are there on an object
hovering in the bus? Since it is not falling,
it must be supplying some kind of upwards force to
balance its downwards gravitational force, and
that's it. Nothing is touching the object, so
nothing can apply a normal force to it. When
the bus moves, the object will be "left behind" by
the bus and hit the back window.
The extra part about closing the windows is an
interesting addition. You can think of air
molecules in the bus like objects, too. But these
are very small and moving very fast, so they're
constantly zipping around and smashing into each
From the point of view of an air molecule, the
school bus moves really slowly, so they will
hardly notice while they are smashing into each
other. If the windows are closed, the air
molecules will still fill the inside of the bus
pretty evenly. They will bump into the hovering
object almost evenly from all sides, so there
will not be a net force from the air.
If the windows are open, a current of air might
start to flow from the front windows towards the
back windows. Now, more air molecules will bump
into the hovering object from the front of the
bus. This will cause a net force that we call
drag, and push the hovering object to the back
of the bus even faster.
That depends on how it's hovering and how much air
resistance it has. If it's something that isn't
falling to the floor of the school bus because the
air is keeping it aloft, then it will move along
with the bus because the air that is holding it is
moving along with the bus. If you have a toy
helicopter or something like that, however, that
is staying aloft because it has an engine that is
giving it lift, then it will hit the back window.
If the windows are open, then the air in the bus
will be able to flow backwards, and it depends on
how quickly the air inside can move.
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