Inertia problem? Inertia of air?

Inertia for air molecules is no different from anything solid such as the metal of the bus. What makes the difference is momentum which equals mass times velocity. The mass of air is small compared to metal.

I agree with @mattbrowne that air is subject to inertia like everything else with mass. This is readily demonstrated with a helium balloon in a car. Apply the brakes and the balloon will paradoxically move backward because air inside the vehicle is temporarily thrown forward, piling up and increasing its density at the windshield, sending the balloon toward the rear where the air is temporarily thinner. Sort of like horizontal buoyancy. The opposite will happen during acceleration, and likewise the balloon moves toward the inside of a curve.

Both the fly (which is denser than air) and the local mass of air in which it is flying should be affected the same way by inertia as you are. The fly, however, has evolved the ability to hover based on input from its visual system, compensating for wind and gusts of air – or in this case small movements of air in a bus. So it can stay in the same place relative to you and your seat.

By the same token, what happens to a canary in a birdcage on an elevator floor while the elevator either ascends or descends? Can the canary fly around his cage as he does normally or will he be pressed either to the roof or floor of the cage?

@gailcalled Assuming the elevator moves at a constant velocity, nothing of note happens to the canary during the main part of the ride. It’s only during the acceleration at the beginning and deceleration at the end that the canary will experience forces. It will experience a downward force during acceleration of upward motion or deceleration of downward motion, and an upward force during acceleration of downward motion or deceleration of upward motion. Whether these forces are strong enough to inhibit its ability to fly depend entirely on the magnitude of acceleration.

@Mariah Is spot on. And remember that even a fly has sensory organs and a rudimentary brain. The animal is also equipped with wings to fly, and an internal guidance system capable of discerning its surroundings and adjusting its flight patter rapidly. If you have ever attempted to reach out open-handed and grab a hovering fly from the air, you know just how fast the little bugger can react.

Air has inertia, but it is so small as to be not noticeable in small volumes. The first human powered plane behaved more sluggish than calculated because they omitted the mass of air in the calculations. This plane had a large volume but small structure mass. Once they put mass of air into the calculations, predictions matched performance.