Is the gas density gradient really so significant that it allows a balloon to move towards the front of the car? Any way we can add some calculations to this?
The density gradient is not really that important. You could do this with a car filled with water and a cork instead of a balloon, and the effect would be the same, even though the density gradient in water would be pretty darn close to zero.
What matters is that the direction of the buoyant force is precisely opposite the direction of the overall force acting on the fluid (because that's what keeps the hypothetical fluid that your balloon displaces from moving due to said overall force). Importantly, this overall force needs to be measured in the reference frame of the fluid, which in this case is accelerating.
OK, so let's draw out a force diagram on the air inside the car. There's a force upward from the floor, exactly balanced by the force of gravity down. There's similarly a force from the back of the car , which is what's accelerating the air, from the point of view of a non-accelerating observer. From the point of view of an observer in the car, however, the air is _not_ accelerating, but the force from the back of the car is still there. So there must be a force backwards on the air, to balance the force of the back of the car. You can call it "inertial force" or "gravity" (in the general-relativistic sense) but the upshot is the same: the overall force on the air is down and backwards, so the buoyant force is up and forward, along the same line. For a helium balloon the buoyant force is stronger than "gravity", so it goes up and forward.
bzbarsky|7 years ago
What matters is that the direction of the buoyant force is precisely opposite the direction of the overall force acting on the fluid (because that's what keeps the hypothetical fluid that your balloon displaces from moving due to said overall force). Importantly, this overall force needs to be measured in the reference frame of the fluid, which in this case is accelerating.
OK, so let's draw out a force diagram on the air inside the car. There's a force upward from the floor, exactly balanced by the force of gravity down. There's similarly a force from the back of the car , which is what's accelerating the air, from the point of view of a non-accelerating observer. From the point of view of an observer in the car, however, the air is _not_ accelerating, but the force from the back of the car is still there. So there must be a force backwards on the air, to balance the force of the back of the car. You can call it "inertial force" or "gravity" (in the general-relativistic sense) but the upshot is the same: the overall force on the air is down and backwards, so the buoyant force is up and forward, along the same line. For a helium balloon the buoyant force is stronger than "gravity", so it goes up and forward.
FakeComments|7 years ago
https://www.youtube.com/watch?v=y8mzDvpKzfY