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September 30, 2024
Jet on a Conveyor Belt
Jet on a Conveyor Belt
The airplane on the conveyor problem has been around for a while. It keeps popping up in my social media, though. I decided to approach it from a slightly different place, emphasizing the actual performance of the airplane.
The question is poorly-worded. (Physics questions on the Internet are often poorly-worded, to create attention as commenters argue with each other.) No frame of reference is provided for the "speed of the wheels." Nor is the point on the wheel specified. Comic artist and uber-geek Randall Munroe (xkcd) has explored the possibilities. The only scenario which makes physical sense is where the wheel speed is measured from its center of mass relative to Earth, and the conveyor speed is measured in the same way.
Spoiler
The airplane can take off. As it does, the wheels will be spinning at twice their normal rate, which could create problems if engineering specifications are exceeded.
Knowns
A force diagram for the airplane on the runway is shown below.
It is possible that both drag and lift will be greater than zero at rest, if the airplane is pointed into the wind (as is normal for takeoff). Friction between the wheels and the runway arises from rolling friction and the need to increase the rotational speed of the wheels.
Internet searches yield the following typical data for a 747.
airplane weight = 800 kilopounds = 3.56 x 106 N
airplane mass = 3.63 x 105 kg
airplane thrust = 60 kilopounds per engine(4 engines) = 240 kilopounds = 1.07 x 106 N
number of wheels = 18
wheel radius = 0.62 m
wheel mass = 184 kg
wheel moment of inertia = 46 kg•m2
wheel effective coefficient of rolling friction = 0.02 (on concrete)
Using this information, we can explore the dynamics of takeoff.
Calculations
The average acceleration of the airplane during takeoff can be estimated from this video as 1.8 m/s2, from zero to 90 m/s in 50 seconds. Nominally, the maximum acceleration would be 2.95 m/s2 (=thrust/mass, excluding resistive forces).
The frictional force required to create angular acceleration in the wheels is quite small in relation to the thrust of the engines. Even if this number is doubled by the motion of the conveyor, it is still less than one percent of the engine thrust.
Due to the significant weight of the airplane, the rolling friction of the wheels creates a much greater resistive force, perhaps 7% of the engine thrust.
Rolling friction is quite variable depending upon runway surface and total airplane weight. It will be diminished by lift from the wings, which causes normal force to be less than airplane weight.
The original question asks, can the airplane take off? We conclude that the airplane has the net force needed to accelerate to takeoff speed in spite of the conveyor, so long as rolling friction on the conveyor is not excessive. If somehow the conveyor exerted greater resistive forces on the airplane—if its design were more exotic than what we are led to believe—the airplane could of course be prevented from taking off.
