Can a 747 take off from a conveyor belt runway?

This question is currently going round on Facebook; one of my friends posted it, I answered it, and a few hours later entered a lengthy and circular debate in the comments section.  Here, with more space than a Facebook comments section, I'd like to pose the question, answer it and then address some of the misconceptions.

Here's the question:  Imagine a 747 is sitting on a conveyor belt, as wide and long as a runway.  The conveyor belt is design to exactly match the speed of the wheels, moving in the opposite drection.  Can the plane take off?

Shared originally by Aviwxchasers.Com a news and media site from the US.

My friend shared the post, and answered, "No - there's no way to generate lift."
My original answer:  

"I think you're assuming that the 747 will be moving because its wheels are being driven. The forward thrust of the aircraft comes from its engines, not from making its wheels turn (like a car). So the engines push the aircraft forward and the wheels just slide and skid along the conveyor belt , while the engines push the aircraft to take-off speed. The wheels don't have to turn to allow the aircraft to move."

However, this wasn't deemed sufficient by some other commentators, who (to summarise) posted the following questions or objections.

1. The conveyer would counteract the forward movement produced by the engines' thrust.  The conveyor matches the speed of the wheels in the opposite direction, so there can be no forward motion.
2. Lift is created by air flow over the wings. It doesn't matter how much thrust you have , if doesn't generate airflow over the wing, it won't fly.
3. The method of propulsion should be irrelevant. If the speed of the conveyor matches the speed of the wheels in the opposite direction, there can be no forward motion.
4. Take the vehicle out of the equation, it's all about the wheels and the conveyor. The faster the wheels turn, the faster the conveyor goes. Stick anything you want on top of the wheels, the principles are the same. Newton's third law
5. Indeed it's not about the wheels it's about forward movement (thrust from the engine) needed to take off but that forward movement is being counteracted by the conveyer    

Here, I propose to look at these arguments individually and collectively, and explain why the points which are raised aren't sufficient to stop the aircraft from moving and taking off.

1.  The conveyor would counteract the forward movement produced by the engines' thrust.  The conveyoy matches the speed of the wheels in the opposite direction, so there can be no forward motion.

1A.  The conveyor will only stop the wheels' rotation from generating forward movement.  However, it is not the wheels which are being driven - this is not a car or a truck.  The forward movement of the plane is not a result of the wheels successfully pushing against the conveyor belt; the forward movement of the plane comes from the push of the engines.  The result is that the plane will move forwards (the engine pushes against the air flowing through it, the air pushes back - Newton's Third Law) and the wheels will spin and skid down the track.

It's not "The plane moves forwards because the wheels go round", it's, "The wheels go round because the plane goes forwards [over a surface which has sufficient friction] ."


2. Lift is created by air flow over the wings. It doesn't matter how much thrust you have , if doesn't generate airflow over the wing, it won't fly.

True.  But there is thrust, and it is generating airflow over the wing.  The conveyor belt does not have the ability to resist the movement of the plane, only to counteract the turning of the wheels.  

The plane can move forwards -even along the ground - without its wheels turning.  The engine isn't driving the wheels.

3.  The method of propulsion should be irrelevant. If the speed of the conveyor matches the speed of the wheels in the opposite direction, there can be no forward motion.
and 
4. Take the vehicle out of the equation, it's all about the wheels and the conveyor. The faster the wheels turn, the faster the conveyor goes. Stick anything you want on top of the wheels, the principles are the same. Newton's third law

3A and 4A.  Comment 3 was a response to my question, "What happens if we replace the 747 with a space rocket, aligned horizontally on the conveyor belt runway, on wheels?"  and the comment suggests a misunderstanding about what's causing the forward motion of the plane.  The thrust of a space rocket is completely independent of the any wheels (they normally fly fine without them) and the wheels will just get dragged along the conveyor belt, without turning. 

Yes, the faster the wheels turn, the faster the conveyor belt goes.  But the wheels don't have to turn for the plane to move (as I said in response to 2).  The conveyor belt does not have some property which prevents something from skidding along it, just from preventing any forward motion due to wheels turning on it. 

You can only take the vehicle out of the equation when you realise that the vehicle isn't a car, with the limitations that a car has.

Newton's third law applies to rotating wheels and conveyor belts.  It also applies to the aircraft engines and the air flowing through them, or to space rockets and the fuel burning inside them.  I wonder if the Starship Enterprise (on wheels) would have this problem?

5.  It's all about forward movement (thrust from the engine) needed to take off but that forward movement is being counteracted by the conveyer  

The forward movement is not counteracted by the conveyor.  The conveyor just stops rotating wheels from generating any forward movement.  But if the wheels aren't rotating (because they're not being driven, such as in a car) then there's no movement to resist.

In conclusion, I'd like to offer this video from "Mythbusters" which shows a light aircraft attempting to take off against a conveyor belt (which in this case is being pulled by a pickup truck to match the plane's speed).  To quote one of the engineers, "People just can't wrap their heads around the fact that the plane's engine drives the propellor, not the wheels."

Some other 'everyday maths' articles I've written:

A spreadsheet solution - the nearest point to the Red Arrows' flight path from my house
The Twelve Days of Christmas - summing triangle and square numbers
Why are manhole lids usually circular?
"BODMAS" puzzles - what's the fuss?

Maths Puzzle: How far does the fly fly?

Two boys set off on their bikes, at the same time, to meet each other.  They are 24 miles apart, and the road between them is perfectly straight.  The first boy cycles at 6 mph (he's only a very small boy), and the second boy cycles at 4 mph (because he's even smaller, and can't cycle as quickly).  A fly sets off from the first boy's handlebars at the same time as the boy starts cycling.  It flies in a straight line from the first boy's handlebars until it reaches the second boy's handlebars. It then turns around, and flies back to the first boy's bike, then when it reaches the first boy, it turns around again and back to the second boy's bike, and so on until the two boys meet.  The fly travels at 12 mph.

How far does the fly travel in total, from the moment the boys set off, until the moment they meet?  The time taken for the fly to turn around every time it reaches a boy can be ignored.

I like this type of puzzle - I liked it even more when I spotted an easy way of solving it. 

But first, the long way.

The fly sets off at 12 mph, at the same time as the second boy starts cycling towards it, at 4 mph.  There are 24 miles between them at this point, and the two travellers are approaching at 16 mph.  It will therefore take them


time = distance / speed = 24/16 = 1.5 hours


to meet.  This is the longest single part of the fly's journey, as the boys were at their furthest from each other.


However, in that time, the second boy has cycled 1.50 hrs x 4 mph = 6 miles, and the first boy has cycled 1.5hrs x 6 mph = 9 miles.
The fly has travelled 1.5 hours x 12 mph = 18 miles.  This makes sense - the second boy has cycled 6 miles, and the fly has travelled 18 miles, and 6 + 18 = 24 which was the starting distance between them.


Finally, while the fly has been flying the first leg of its journey, the two boys have reduced the distance between them from 40 miles to 24 - (9 + 6 miles) = 9 miles.

Now, the fly turns around, and starts to fly towards the first boy.  The first boy, remember, is cycling at 6 mph; the fly is still going at 12 mph, so their closing speed is 18 mph.  They have to travel 9 miles (the distance now remaining), so this will take:


time = distance / speed = 9 miles / 18 mph = 0.5 hours (which is 30 minutes).


In that time, the first boy travels 0.5 hours x 6 mph = 3 miles.
The second boy travels 0.5 hours x 4 mph = 2 miles.
And the fly travels 0.5 hours x 12 mph = 6 miles.


This makes sense - note that the first boy and the fly have together covered the nine remaining miles between them, 3 + 6 = 9


So far, the fly has travelled a total of 18 miles + 6 miles = 24 miles.


The distance between the two cyclists is now down to
24 - ((9 + 6) + (3 + 2)) = 4 miles

Continuing for a third leg, our tireless fly starts back from the first boy to the second boy.  The second boy is cycling at 4 mph, the fly is travelling at 12 mph.  Closing speed is 16 mph, and distance to cover is just 4 miles.


4 / 16 = 0.25 hours (15 minutes).


First boy covers 0.25 hours x 6 mph = 1.5 miles
Second boy covers 0.25 hours x 4 mph = 1 mile
Fly travels 0.25 hours x 12 mph = 3 miles


Distance remaining is 24 - (( 9 + 6 ) + (3 + 2) + (1.5 + 1)) = 1.5 miles
Fly has now travelled 18 + 6 + 3 = 27 miles

Clearly, this is going to take a long time to solve through this method.

Here's the shorter way.

The boys have to cover 24 miles.  The first boy cycles at 6 mph, and the second boy at 4 mph.  This means that they are approaching each other with a closing speed of 10 mph, and it will take them 24 miles / 10 mph = 2.4 hours (2 hours and 24 minutes) to complete their journey and meet up. Obviously, there are many different versions of this story, involving trains and so on, but the principle is the same (and you can change the numbers to make them more realistic - I think my boys are pedalling at walking speed!).


The fly, meanwhile, is flying at 12 mph.  This means that in the 2.4 hours it takes the boys to meet, it will fly 2.4 hrs x 12 mph = 28.8 miles.


It really is that easy.  No diagrams, no long complicated adding up then adding up some more.  Sometimes, all that's needed to solve a problem is a different perspective!

If you enjoyed this article, you may like some of my other, more recent posts on puzzles and games that can be solved with a calculator:

Snakes and Ladders (Collatz Conjecture)
Crafty Calculator Calculations (numerical anagrams with five digits)
More Multiplications (numerical anagrams, four digits)
Over and Out (reduce large numbers to zero as rapidly as possible)
Calculator Games: Front to Back
Calculator Games: Up, up and away with Ulam sequences
Calculator Games: The Kaprekar Constant