A hummingbird is hovering in an airplane that is grounded and not moving. The plane then accelerates and takes off. With respect to the plane, what would the bird's motion be?

I've seen this problem proposed in the new scientist a few times. The concensus is that the air (molecules) have little mass and would accelerate freely with the air in the plane. As the air is moving at the same rate as the plane, the humming bird would not notice any change and would remain hovering in a stationary position. Same as when there is a fly in your car, it can remain hovering when you accelerate. However there must be a rate at which the acceleration must be too much. A lot to think about really. I've often thought what would happen if you were in a falling lift and you managed to jump just prior to the lift hitting the floor. Would you get squashed?

The situation is that the plane is stationary, the air inside it is stationary, and the bird is hovering. As the plane starts moving, it will set the air in motion also. But will this motion be imparted to the bird ? I dont think so. The bird is likely to hit the airplane. On the other hand if the plane has been in steady motion for some time and the bird starts to hover, it already has the velocity of the plane so it wont hit the airplane walls.

I'm glad to know I am not the only person who thinks about what they would do if a lift failed!

Unlike a fly, a humming bird has a small but considerable mass. Thus it will be affected during the acceleration phase of the airplane (just like any other passenger on board) and will drift towards the back of the plane. Most likely, the bird will correct for this motion, so you would have to blind-fold it to really test it out. Surrounding air molecules cannot compensate for this effect. I am sure if you took a small helium filled balloon inside an airplane that was neutrally buoyant, you would notice the drift during the acceleration phase.

Regarding the crashing elevator:

First of all, elevators are highly unlikely to fail. As a matter of fact, I have never heard of such a case other than in some Hollywood movie. The design safety factor of elevators lies between 6 and 8, I believe. Thus for a 1000 kg rated elevator you can load at least 6000 kg before something like a cable would break. Even if a cable did break, there are braking mechanisms along the guide wheels that will bring the car to a stop, once it exceeds a certain speed.

Let's assume though that both the cable and the braking mechanism failed. An elevator is a high drag aerodynamic vehicle, which will not enter a free fall acceleration. Its terminal speed will be, I assume, lie somewhere between 50 and 80 km/h depending on friction and aero-braking. Needless to say, you cannot compensate such a speed with an upward jump, and even if you could, you would not need to jump in the first place, since the strength in your legs could easily absorb the crash.

Again, in the history of elevators, I have never heard that one failed, but please prove me wrong if you know of such an accident. In the unlikely event that such a failure should ever happen though, the only sensible thing to do would be to lie down flat on the floor of the elevator, perhaps with your head cushioned under your hands. That way when the car does hit the floor, your mass is evenly distributed, and you will be less likely to suffer from internal damage during the sudden deceleration.

Even when lying on the floor, the internal trauma would probably kill you. Most deaths from car crashes are caused by the internal organs continuing on the chosen path when the body has stopped moving. Another interesting conudrum is this: If a van was full thousands of humming birds (say 200Kg) and they are all hovering, what is the vans total mass?

You mean "what is the vans total weight?"

Thanks for the advise Axel!!

Your legs are much less likely to break if you lie down rather than stand up. The same is true for your internal organs 'connected' inside your torso. Why do you think astronauts returning from the moon were lying inside their capsules rather than sitting or standing. It is the only way to sustain high deceleration without suffering any damage to the external (bones) or internal (organs) structure of your body. Of course there is a limit to how much deceleration a body will take even in a lie-down position (ca. 20 g's non-sustained I have been told).

The mass of the van with the birds inside is constant (Mtotal = Mvan + 200kg) whether the birds are hovering inside the van or not. The same is true for the weight of the van, since the momentum imparted by the birds on the air is absorbed by the interior of the van which expresses itself in a down force equal to the weight of the birds. - If you study CFD you should be familiar with the law of conservation of momentum and how to apply it to a control volume. Ditto for conservation of mass!

... to the external (bones) ... structure of your body. Are you talking about ants ?

Perhaps not carefully worded...I meant to say the bone structure that protects internal organs and the brain as well as nerves running through the spinal chord. From an 'organ point of view' I consider these bones 'external'. But please excuse my anatomical inaptitude - I am a Ph.D. not an M.D...

Regarding the bird (or a fly) in a plane, here are my thoughts:

Newton's laws must be observed i.e. the bird has mass and if it is to accelerate then a force must be acting on the bird. If the bird is in the air then the force can only come from air pressure differences across the bird's body. If you integrate the static pressure across the hovering bird's body then you must get m*g, where m is the mass of the bird and g is 9.81m/s^2.

If the bird is to remain stationary (relative to the plane's frame of reference) it must overcome the combined force of the bird's weight and the plane's acceleration by "hovering" at an angle determined by the vector sum of the m*g and m*a where a is the acceleration of the plane. This is of course only if the bird to remain stationary for some reason.

I realize this sounds counter intuitive - if you put the bird next to the plane (not near the engine…) and assuming the bird was trying to keep up, there is no way that a bird could accelerate as fast as a plane! Why? Because of the enormous amount of drag to overcome. However, inside the plane, this drag is overcome by powerful jet engines and so the scenario is very different.

Another way of thinking is to consider holding out a 1kg ball on a string from your seat, ignoring the motion of the air, there will be sqrt(g^2+a^2)N tension in the string. Now theoretically if you strap a miniature rocket onto the ball and produce sqrt(g^2+a^2)N of force then the ball will hover stationary relative to the moving plane.

Regarding the air inside the plane accelerating, the situation is like a piston acting on a column of air. The information would travel from the back of the plane via pressure waves (mainly) and partly from the sides walls through shear.

axel rohde got it right in his posts. however, michael's analysis is the most detailed scenario, and he has nailed all the subtleties of the situation.

the air inside the cabin is indeed accelerated by pressure waves arising at the rear wall and/or floor. as michael points out, if the bird were outside the plane, it would be unable to keep up with the plane. however, when it is inside, it would be subject to a flowfield with unsteady motion of air in the direction of travel of the plane (visualize in the ground frame). this would accelerate the bird somewhat in the direction of the plane's travel. the bird would actually benefit from spreading its wings in the plane perpendicular to the direction of motion (neglecting gravity). however, this "sweeping away" of the bird, and its own efforts to keep away from walls, will not be sufficient if the plane accelerates quickly enough. then the bird would be thrown against the rear wall / floor, and brought to the speed of the plane by acceleration by the force of direct contact (or even faster, if the bird bounces).

this tells us that birds should fasten their seat belts once the plane is in motion, just like the rest of us. on a side note, switching from humming to mocking, what would you have if someone one gave you 2000 mockingbirds?

answer: two kilo mockingbird