Science Puzzles

Fun and Games

Science Puzzles: Origins

There are puzzles of a scientific often posted on social media. People often form very different views on the correct solution. Those who give the correct answer do not give a full scientific justification. Those who give the wrong answer have made assumptions that are not true.

Aircraft on Conveyor Belt

This puzzle concerns a 474 aircraft that is standing on a conveyor belt that moves backward at the same speed as the aircraft’s wheels rotate so that the aircraft stays in the same place while the engines are running.

The question is will the aircraft take off?

Aircraft Specification and assumptions

SI units will be used. These can easily be converted to other units online. The 747 aircraft specification rounded for convenience.

The maximum laden mass is \(m = 400,000kg\).
It has four engines each producing a thrust of \(250kN\) totalling \(F = 1,000kN\).
Its takeoff velocity is \(v_t = 80ms^{-1}\).

Some reasonable assumptions will be made. Any of these can be added to the calculations if required.

Drag, which which is a function of air speed, will be ignored for takeoff.
Wheel to runway friction, which is a function og ground speed will be ignored.
The decrease in aircraft mass due to burning fuel will be ignored.
There is no wind.
The airport is situated on the Equator at sea level.
The aircraft is on the West end of a "0" runway heading East.

Normal Takeoff

First do the calculation with the 747 taking off normally without the conveyor. An aircraft has two definitions of speed.

Air speed is the speed the aircraft is travelling through the air.
Ground speed is the speed that the aircraft travels across the ground below it.

Air speed and ground speed are usually different during flight.

Using Newton’s Second Law, calculate the acceleration of the aircraft.

\[a = \frac{F}{m} = \frac{1.0\cdot 10^6}{4.0\cdot 10^5} = 2.5 ms^{-2}\]

Use the equation of motion to determine the time from starting at rest until takeoff.

\[v_t = u_t + at \\ 80 = 0 + 2.5t \\ t = 32s\]

During takeoff, the air speed and ground speed are the same.

Conveyor Takeoff

When the aircraft attempts to take off from a moving conveyor many people assume that the airspeed and ground speed are the same as for a normal takeoff. If this were the case then there would be no lift and it would not take off. Also note that the conveyor must increase its speed to match the wheel speed otherwise the aircraft would fall off the conveyor.

The assumption that the air speed and ground speed are the same during conveyor takeoff is not true.

When the aircraft starts its engines they generate thrust. This causes the aircraft to accelerate.

The aircraft wheels are not powered by the engine, they rotate because of friction with the ground.
The fact that the wheel turn proves that the plane isn’t stationary.
The conveyor cancels out the ground speed acceleration fixing the effective ground speed to zero.
The engines apply thrust to the air increasing lift.
The aircraft’s air speed accelerates at the same rate as for normal takeoff.

The aircraft will always take off!