Re: Mythbusters, Plane On A Treadmill Disscussion
This is a fairly simple physics problem if you know what the important factors are...
For one, you need to understand what a force vector is. A force vector is a force that has both a maganitude and a direction.
When you have 2 seperate forces you add them together and come up with a resultant force vector.. like the following:
http://users.iafrica.com/m/ms/mshenker/assets/images/Vector_402.jpg
Fv is the force pulling upwards. Fh is the force pulling forward and Fr is the resultant force vector. Fr indicates the direction that an object , if forces v and h were both acting on it, would be accelerated towards.
It might be obvious but please notice that we are not adding speeds here. We are simply looking at two distinct forces acting on the object and coming up with thier net result. Once you have that net result you can determine the rate of acceleration and thus the speed of the object at any point in time.
Ok, so in the problem of the plane on a treadmill we have an object. The object happens to be a jet plane but that's not really all that important. The essential question is, given two forces acting on the object, can the object move and if so, which direction?
To understand the forces acting on the plane we need to know a couple of things about it. One is the jet engine.
The jet engine works by accelerating air in one direction and thus pushing the plane in the other direction. Newton 's law. Every action has an equal and opposite reaction. Jet engine blows air out in one direction, the engine pushes the plane in the other. Simple enough.
The force from the Jet engine is considerable. Just for the sake of specifics lets use an F-16.
The engine in an F-16 can provide 27,000 pounds of thrust.
So lets call the force from the engine 'F1'.
F1 = 27,000 lbs
What other forces are acting on the plane?
Well there are probably a few such as gravity, but the only one we are concerned with is the force being applied by the treadmill.
Since the wheels on the plane are "free rolling" the only force that the treadmill can apply is through friction in the bearings themselves and through the 'moment of intertia' in the wheels. The 'moment of intertia' refers to the force it takes to spin the wheels faster and faster. If you have ever put a bicycle upside-down and tried to pedal it as fast as you could you know what this force is.
Once you get the wheel spinning really fast it gets harder and harder to make the wheel spin any faster, even though there is nothing slowing it down directly. We already know that the jet has no trouble at all moving along the runway and spinning those wheels at a couple hundred miles an hour. So we can deduce that this force is quite small relativly speaking. Just for arguements sake let's say that it requires 500 lbs of thrust to over come that force under normal circumstances. I think that is probably way way too high, but it's just to assign it a meaningful number. The friction from the wheel bearings is going to be very small. That is thier purpose afterall, to reduce rolling friction. We can practically ignore any friction due to the bearings. But just to be consistant let's give them another 500 lbs of thrust, which frankly is ridiculously high and for that to be right the bearings would need to be filled with sand.
However it will help us prove our point.
Let's call the sum of those two forces F2. 1000 lbs.
So in the problem it states that the treadmill matches the planes speed in the reverse direction. What does this mean?
Well, when the plane is stationary the treadmill is also tationary. When the plane begins to push forward the treadmill spins up. The result is the plane moving forward at say 10mph relative to the air-traffic control tower, the treadmill spinning at 10 mph in the opposite direction and the wheels on the plane spinning at the equivalent of 20mph.
Now extrapolate this forward. as the plane accelerates the wheels accelerate at double thier normal rate. We have in essence already answered the question of "will the plane move?". Obviosuly at low speeds it moves forward at virtually the same rate it did before.
Remember, 27,000 lbs of thrust is alot compared to the tiny amount of force that is required to spin a wheel at 20mph.
essentially what we have is this: F1 > F2.
Since F1 is bigger than F2, we know that the object will be accelerated in the direction of F1. Meaning it will infact move forward.
I don't have the data available here for me to actually calculate the amount of force it would take to spin those wheels up to double the take off speed of the F-16. I don't imagine it would be anywhere near enough to totally counteract the enormous amount of thrust applied by the jet engine. So in short, yes the plane takes off.
This is a fairly simple physics problem if you know what the important factors are...
For one, you need to understand what a force vector is. A force vector is a force that has both a maganitude and a direction.
When you have 2 seperate forces you add them together and come up with a resultant force vector.. like the following:
http://users.iafrica.com/m/ms/mshenker/assets/images/Vector_402.jpg
Fv is the force pulling upwards. Fh is the force pulling forward and Fr is the resultant force vector. Fr indicates the direction that an object , if forces v and h were both acting on it, would be accelerated towards.
It might be obvious but please notice that we are not adding speeds here. We are simply looking at two distinct forces acting on the object and coming up with thier net result. Once you have that net result you can determine the rate of acceleration and thus the speed of the object at any point in time.
Ok, so in the problem of the plane on a treadmill we have an object. The object happens to be a jet plane but that's not really all that important. The essential question is, given two forces acting on the object, can the object move and if so, which direction?
To understand the forces acting on the plane we need to know a couple of things about it. One is the jet engine.
The jet engine works by accelerating air in one direction and thus pushing the plane in the other direction. Newton 's law. Every action has an equal and opposite reaction. Jet engine blows air out in one direction, the engine pushes the plane in the other. Simple enough.
The force from the Jet engine is considerable. Just for the sake of specifics lets use an F-16.
The engine in an F-16 can provide 27,000 pounds of thrust.
So lets call the force from the engine 'F1'.
F1 = 27,000 lbs
What other forces are acting on the plane?
Well there are probably a few such as gravity, but the only one we are concerned with is the force being applied by the treadmill.
Since the wheels on the plane are "free rolling" the only force that the treadmill can apply is through friction in the bearings themselves and through the 'moment of intertia' in the wheels. The 'moment of intertia' refers to the force it takes to spin the wheels faster and faster. If you have ever put a bicycle upside-down and tried to pedal it as fast as you could you know what this force is.
Once you get the wheel spinning really fast it gets harder and harder to make the wheel spin any faster, even though there is nothing slowing it down directly. We already know that the jet has no trouble at all moving along the runway and spinning those wheels at a couple hundred miles an hour. So we can deduce that this force is quite small relativly speaking. Just for arguements sake let's say that it requires 500 lbs of thrust to over come that force under normal circumstances. I think that is probably way way too high, but it's just to assign it a meaningful number. The friction from the wheel bearings is going to be very small. That is thier purpose afterall, to reduce rolling friction. We can practically ignore any friction due to the bearings. But just to be consistant let's give them another 500 lbs of thrust, which frankly is ridiculously high and for that to be right the bearings would need to be filled with sand.
However it will help us prove our point.
Let's call the sum of those two forces F2. 1000 lbs.
So in the problem it states that the treadmill matches the planes speed in the reverse direction. What does this mean?
Well, when the plane is stationary the treadmill is also tationary. When the plane begins to push forward the treadmill spins up. The result is the plane moving forward at say 10mph relative to the air-traffic control tower, the treadmill spinning at 10 mph in the opposite direction and the wheels on the plane spinning at the equivalent of 20mph.
Now extrapolate this forward. as the plane accelerates the wheels accelerate at double thier normal rate. We have in essence already answered the question of "will the plane move?". Obviosuly at low speeds it moves forward at virtually the same rate it did before.
Remember, 27,000 lbs of thrust is alot compared to the tiny amount of force that is required to spin a wheel at 20mph.
essentially what we have is this: F1 > F2.
Since F1 is bigger than F2, we know that the object will be accelerated in the direction of F1. Meaning it will infact move forward.
I don't have the data available here for me to actually calculate the amount of force it would take to spin those wheels up to double the take off speed of the F-16. I don't imagine it would be anywhere near enough to totally counteract the enormous amount of thrust applied by the jet engine. So in short, yes the plane takes off.
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would the plane be able to travel forward and take off? The answer to that question would certainly be no, although impossible to prove.
